MATHEMATICS

ACCOUNTING PROFESSION OPTION

For Rwandan Schools

Senior
5
for Rwandan SchoolsStudent Book

EXPERIMENTAL VERSION
 © 2023 Rwanda Basic Education Board
All rights reserved
This book is the property of the Government of Rwanda.
Credit must be provided to REB when the content is quoted
FOREWORD
Dear Student,
Rwanda Basic Education Board (REB) is honoured to present Senior 5
Mathematics book for the students of Accounting Profession Option which
serves as a guide to competence-based teaching and learning to ensure
consistency and coherence in the learning of the Mathematics. The Rwandan
educational philosophy is to ensure that you achieve full potential at every level
of education which will prepare you to be well integrated in society and exploit
employment opportunities.
The government of Rwanda emphasizes the importance of aligning teaching
and learning materials with the syllabus to facilitate your learning process.
Many factors influence what you learn, how well you learn and the competences
you acquire. Those factors include the relevance of the specific content, the
quality of teachers’ pedagogical approaches, the assessment strategies and
the instructional materials available. In this book, we paid special attention
to the activities that facilitate the learning process in which you can develop
your ideas and make new discoveries during concrete activities carried out
individually or in groups.
In competence-based curriculum, learning is considered as a process of active
building and developing knowledge and meanings by the learner where
concepts are mainly introduced by an activity, situation or scenario that helps
the learner to construct knowledge, develop skills and acquire positive attitudes
and values.
For efficiency use of this textbook, your role is to:
– Work on given activities which lead to the development of skills;
– Share relevant information with other learners through presentations,
discussions, group work and other active learning techniques such
as role play, case studies, investigation and research in the library, on
internet or outside;
– Participate and take responsibility for your own learning;
– Draw conclusions based on the findings from the learning activities.
To facilitate you in doing activities, the content of this book is self-explanatory
so that you can easily use it yourself, acquire and assess your competences. The
book is made of units as presented in the syllabus. Each unit has the following
structure: the unit title and key unit competence are given and they are followed
by the introductory activity before the development of mathematical concepts
that are connected to real world problems more especially to production,
finance and economics.

Mathematics | Student Book | Senior Five | Experimental Version i

The development of each concept has the following points:
• Learning activity which is a well set and simple activity to be done by
students in order to generate the concept to be learnt;
• Main elements of the content to be emphasized;
• Worked examples; and
• Application activities to be done by the user to consolidate competences or
to assess the achievement of objectives.
Even though the book has some worked examples, you will succeed on the
application activities depending on your ways of reading, questioning, thinking
and handling calculations problems not by searching for similar-looking
worked out examples.
Furthermore, to succeed in Mathematics, you are asked to keep trying;
sometimes you will find concepts that need to be worked at before you
completely understand. The only way to really grasp such a concept is to think
about it and workrelated problems found in other reference books.
I wish to sincerely express my appreciation to the people who contributed
towards the development of this book, particularly, REB staff, development
partners, Universities Lecturers and secondary school teachers for their
technical support. A word of gratitude goes to Secondary Schools Head Teachers,
Administration of different Universities (Public and Private Universities) and
development partners who availed their staff for various activities.
Any comment or contribution for the improvement of this textbook for the next
edition is welcome.

Dr. MBARUSHIMANA Nelson

Director General, REB.

ii Mathematics | Student Book | Senior Five | Experimental Version

ACKNOWLEDGEMENT
I wish to express my appreciation to the people who played a major role in the
development of this Mathematics book for Senior Five students in Accounting
Profession Option. It would not have been successful without active participation
of different education stakeholders.
I owe gratitude to different universities and schools in Rwanda that allowed
their staff to work with REB in the in-house textbooks production initiative.
I wish to extend my sincere gratitude to Universities Lecturers, Secondary
school teachers and staff from different education parterres whose efforts
during writing exercise of this book were very much valuable.
Finally, my word of gratitude goes to the Rwanda Basic Education Board staffs
who were involved in the whole process of in-house textbook Elaboration.

MURUNGI Joan

Head of CTLR Department

Mathematics | Student Book | Senior Five | Experimental Version iii

TABLE OF CONTENT
FOREWORD........................................................................................................... i
ACKNOWLEDGEMENT.....................................................................................iii
Unit 1: Matrices and determinants................................................................................ 1
1.1 Generalities on matrices..........................................................................................1

1.2. Operations on matrices...........................................................................................6

1.3. Determinants of square matrices.....................................................................18

1.4. Finding the inverse and solving simultaneous linear equations........26

End of unit assessment 1..............................................................................................37

Unit 2: Differentiation/Derivatives...............................................................................40
2.1 Differentiation from first principles.................................................................41

2.2. Rules for differentiation.......................................................................................44

2.3. Some applications of derivatives in Mathematics.....................................56

End of unit assessment 2..............................................................................................60

Unit 3: Applications of derivatives in Finance and in Economics..............62

3.1. Marginal quantities................................................................................................62

3.2. Minimization and maximization of functions.............................................65

3.3. Price elasticity..........................................................................................................67

End of unit assessment 3..............................................................................................70

Unit 4: Univariate Statistics and Applications......................................................71

4.1 Basic concepts in univariate statistics.............................................................72

4.2 Organizing and graphing data............................................................................83

4.3 Numerical descriptive measures.................................................................... 106

4.5 Measure of symmetry.......................................................................................... 116

4.6 Examples of applications of univariate statistics in mathematical

problems that involve finance, accounting, and economics....................... 121

End of unit assessment 4........................................................................................... 123

iv Mathematics | Student Book | Senior Five | Experimental Version

 Unit 5: Bivariate statistics and Applications.......................................................125
5.1 Introduction to bivariate statistics................................................................ 126

5.2 Measures of linear relationship between two variables: covariance,

Correlation, regression line and analysis, and spearman’s coefficient of
correlation....................................................................................................................... 128

End of unit assessment 5........................................................................................... 140

REFERENCES ................................................................................................ 142

Mathematics | Student Book | Senior Five | Experimental Version v

 Unit
1 Matrices and determinants

Key Unit competence: Use matrices and determinants notations and properties
to solve simple production, financial, economical,
and mathematical related problems.

Introductory activity

The table below shows the revenue and expenses (in Rwandan francs) of
a family over three consecutive months:

October November December

Revenue 450,000 460,000 700,000
Expenses 440,000 295,000 890,000
a) What was the family’s revenue in October?
b) By how much money did the family’s revenue increase from
October to November?

1.1 Generalities on matrices

1.1.1. Definitions and notations

Learning Activity 1.1.1

A shop selling shirts records the number of each type of shirts it sells over a
period of two weeks. In the first week, it sells 12 small size shirts, 8 medium
shirts and 5 large shirts.
In the second week, it sells only 9 small size shirts and 3 medium size
shirts.
a) What are the two criteria the shopkeeper will use to record these
data?
b) Record this information in a rectangular array consisting of double
entries.
c) Such a table is called a “matrix”. Describe the components of a matrix.

Mathematics | Student Book | Senior Five | Experimental Version 1

CONTENT SUMMARY
– A matrix is a rectangular arrangement of numbers, in rows and columns,
within brackets ( ) or [ ] . A matrix is denoted by a capital letter: A, B, C, ….
Rows are counted from the top of the matrix to the bottom of the matrix;
columns are counted from the leftmost side of the matrix to the rightmost side
of the matrix.
– The numbers in the matrix are called entries or elements.
The position of an entry in the matrix is shown by lower subscripts, such as a j
:the entry on the ith row and jth column.
– If matrix A has n rows and p columns, then we say that the matrix A is of
order n × p , read n by p, where the product n × p is the number of entries
in the matrix.
Note: In finding the order of a matrix, we do not perform the multiplication
n × p , we just write n × p , but for finding the number of entries of a matrix
given by its order n × p ,we calculate the product n × p .
If A is a matrix of order n × p , then A can generally be written as A = ( aij ) ,where
i and j are positive integers, and; 1 ≤ i ≤ n ;1 ≤ j ≤ p .
– A matrix with only one row is said to be a row matrix; that is a matrix of
order1× p .

Thus, ( 2 4 7 ) is a row matrix.

A matrix with only one column is said to be a column matrix; that is a matrix
of order n ×1 .

1 
 
Thus,  3  is a column matrix.
6
 
– A square matrix is a matrix in which the number of rows is equal to the
number of columns; that is, matrix A of order n × p is a square matrix if and
only if n = p ;
In this case, instead of saying a matrix of order n × n , we, sometimes, simply say
a matrix of order n .

2 Mathematics | Student Book | Senior Five | Experimental Version

  4 5 2
 3 −2   
Thus, A =
=  ; B  2 0 3  are square matrices of orders 2 and 3,
1 0  1 7 6
 

respectively.

If A = ( aij ) is a square matrix of order 2 , then i and j assume values in the set

a a12 
{1, 2}.Therefore, ( aij ) =  11 .
 a21 a22 
In the same way, if A = ( aij ) is a square matrix of order 3 , then i and j assume

 a11 a12 a13 

 
values in the set{1, 2,3} .Therefore, ( aij ) =  a21 a22 a23  .
a a32 a33 
Example 1.1.1.  31

1 3 4 
 
In the matrix =M (= aij )  5 12 13  ,
7 6 0 
 
a) Write down the value of a23 and the value of a31
b) Explain why the matrix is a square matrix

Solution:
a) The entry on the second row and third column is a23 = 13 ;
The entry on the third row and first column is a31 = 7 .
b) M is a square matrix because it has the same number of rows and
columns

Mathematics | Student Book | Senior Five | Experimental Version 3

 Application activity 1.1.1
1. Write down the order of each of the following matrices:

 8 6 2 7
a) A =   b) B = ( −2 1 3) c) C =  
 −1 1 0   −5 

2. A shoe shop sells shoes for men and ladies. The first week, it sold 7
pairs of men’s shoes and 15 pairs of ladies’ shoes. The second week,
it sold 9 pairs of ladies ‘shoes and 4 pairs of men’s shoes. Record
this information as a 2 × 2 matrix, stating what the rows stand for,
and what the columns stand for.

1.1.2. Equality of matrices

Learning Activity 1.1.2

Consider the following situations:

Situation1:
A class consists of boys and girls who are boarders or day scholars. The
class teacher records the data by the matrix
 8 9
A=  , where the columns represent the numbers of boys and girls,
17 6 
and the rows represent the numbers of boarders and day scholars.

Situation2:
Two brothers sell shirts and shoes, in two different shop I and II, for two
consecutive weeks. The Elder brother records his data by the matrix
 8 9
B=  , where the columns represent the numbers of shirts and
17 6 
shoes, and the rows represent the numbers of items sold in week1, and in
week2.
 8 9
The younger brother, also, records his data by the matrix C =  ,
17 6 
where the columns represent the numbers of shirts and shoes, and the rows
represent the numbers of items sold in week1, and in week2. Comment on
the following, for matrices A, B and C:

4 Mathematics | Student Book | Senior Five | Experimental Version

 a) Number of rows and columns
b) Corresponding entries (that is entries occupying the same
positions)
c) Nature of the elements.
d) Predict which two of the matrices above (A, B and C) are equal.
e) What are the conditions for two matrices to be equal?

CONTENT SUMMARY
A = ( aij )
Two matrices and B = ( bij ) are equal if and only if:
i) they have the same order;
ii) the corresponding entries (that is the entries occupying the same position,
in terms of rows and columns) are equal.
iii) The nature of the entries in the two matrices is the same.
Note: When discussing the equality of matrices, we assume the nature of the
entries in the two matrices to be the same.

 3 0   2 +1 7 − 7 
   
−18  13
Thus, matrices A=  −1 and B =  2  are equal.
 −9   −13 
   8 
 16 23   4
Example1.1.2.

 2x −1 −3   9 −3 
Given that matrices A =   and B =   are equal, find the
values of x and y .  2 2x + y   2 −11

Solution:
Matrices A and B are both of order 2 × 2 and for the corresponding entries to
be equal, we have:

 2 x − 1 =9  x=5
 . Solving simultaneously, we get 
2 x + y = −11  y = −21

Mathematics | Student Book | Senior Five | Experimental Version 5

 Application activity 1.1.2
1. 1. Determine whether the following matrices A and B are equal or
not:

 1 1 5 − 4 3 
 1 1      1 23 − 5 
a) A =   and B = 1 1 b) A = 12  and B =  
 1 1  1 1
 2   9 −1 4 
   3

 2 1
 2x − x  1 y
2. Given that matrices X = 3  and Y =  2  are equal,
  t + t −5 
 0 z 

find the values of x, y, z and t .

1.2. Operations on matrices

1.2.1. Addition and subtraction of matrices

Learning Activity 1.2.1

A retailer sells two products, P and Q, in two shops, S and T.
She recorded the numbers of items sold for the last three weeks in each
shop by the following matrices:

 6 5 13  7 8 2 
S =  and T =  .
11 13 10   7 17 20 

a) Write down the order of each of the two matrices S and T. How are
these two orders?
b) Determine a single matrix for the total sales for this retailer for
the last three weeks in the two shops.
c) Predict the conditions for two matrices to be added and how to
obtain the sum of two matrices.

6 Mathematics | Student Book | Senior Five | Experimental Version

CONTENT SUMMARY
Matrices that have the same order can be added together, or subtracted. The
addition, or subtraction, is performed on each of the corresponding elements.
Thus, if A = ( aij ) and B = ( bij ) are two matrices of the same order n × p , then
the sum of these matrices is the matrix C= A + B , of order n × p ,defined by
A + B = ( aij ) + ( bij ) = ( aij + bij ) , where i and j are positive integers, and
1 ≤ i ≤ n;1 ≤ j ≤ p
In the same way, subtracting matrix B from matrix A yields in matrix D= A − B
, of order n × p ,defined by A − B = ( aij ) − ( bij ) = ( aij − bij ) , where i and j are
positive integers, and 1 ≤ i ≤ n;1 ≤ j ≤ p .

a a12   b11 b12   a11 + b11 a12 + b12 

In particular,  11 + =  , and
 a21 a22   b21 b22   a21 + b21 a22 + b22 

 a11 a12 a13   b11 b12 b13   a11 + b11 a12 + b12 a13 + b13 
     
 a21 a22 a23  +  b21 b22 b23  =
 a21 + b21 a22 + b22 a23 + b23 
a a32 a33   b31 b32 b33   a31 + b31
 a32 + b32 a33 + b33 
 31

Example 1.2.1.

11 29   6 34 
Given matrices P =   and Q =   , find the matrices:
 7 14  3 7 
a) Q − P

b) P + Q

Solution:

 6 34  11 29   6 − 11 34 − 29   −5 5 
a) Q=
−P  − =   =   
 3 7   7 14   3 − 7 7 − 14   −4 −7 

11 29   6 34  11 + 6 29 + 34  17 63 

b) P=
+Q  + =   =   
 7 14   3 7   7 + 3 14 + 7   10 21 

Mathematics | Student Book | Senior Five | Experimental Version 7

 Application activity 1.2.1
1. Say, with reason, whether matrices A and B can be added or not. In
case they can be added, find their sum and the difference A − B

3 5 10 3 5 
a) A =
= ; B  
8 1  11 11 4 

 1 17  11 29 
= b) A = ; B  
14 3  8 6
2. In a sector of a district, there are three secondary schools, A, B and C
having both boarding and day sections for both boys and girls. The
distribution of the students in the three schools are given, respectively

 350 500   420 312  124 0 

=
by the matrices A =  ; B = ;C  ,
 135 248   795 287   41 264 
where the first rows indicate the number of girls, the second rows
the number of boys, the first columns the number of boarders and
the second columns the number of day scholars in the three schools.
The Sector Education Officer (S.E.O) would like to record these data as a
single matrix S.
a) Which operation should he/she perform on the three matrices to
obtain matrix S?
b) Write down matrix S.
c) Use matrix S to answer the following questions:
i) How many day scholars are there from these three schools?
ii) How many girls are boarders from these three schools?

8 Mathematics | Student Book | Senior Five | Experimental Version

 1.2.2. Scalar multiplication

Learning Activity 1.2.2

The monthly rental prices (in thousand Rwandan Francs) of three
apartments without VAT (Value Added Tax) are recorded by the matrix
below:

M = (150 120 300 ) .

a) How do you calculate the VAT on an item?
b) What is the single operation to use in order to obtain the matrix M’
representing the monthly rental prices of the three apartments,
including 18% of VAT?
c) How do you obtain matrix M’?
d) Write down matrix M’

CONTENT SUMMARY
A matrix can be multiplied by a specific number; in this case, each entry of the
matrix is multiplied by the given number. This type of multiplication
is called
scalar multiplication, since the matrix is multiplied by a single real number,
and real numbers are also called scalars.

A = ( aij ) = ( aij )
α A α= (α a ) ,
Thus, if of any order n × p and α a scalar, then ij

where i and j are positive integers, and 1 ≤ i ≤ n;1 ≤ j ≤ p

a a12   α a11 α a12 

In particular, α  11 =  , and
 a21 a22   α a21 α a22 

 a11 a12 a13   α a11 α a12 α a13 

α  a21 a22
 
a23  =  α a21 α a22 α a23 
a a32 a33   α a31 α a32 α a33 
 31

Example 1.2.2.

 1 −2  6 3
Given matrices A =   and B =   , find the matrix 2 A + 3B
7 4  2 7

Mathematics | Student Book | Senior Five | Experimental Version 9

Solution:

 1 −2   6 3   2 −4  18 9   20 5 
2A =
+ 3B 2   + 3 =  + =   
 7 4   2 7  14 8   6 21  20 29 

Application activity 1.2.2

3 4  −2 1 
1. Given matrices A =   and B =   ,find:
5 0  5 3
a) 4 A − 5 B
b) 2( A + B)
3) 2. In November a shop sells clothes and shoes for both boys and
girls.
The number of clothes and shoes sold for boys and girls are

 15 21 
recorded in the matrix N =   , where the number of boys
and girls are in  19 28 

columns, and the numbers of shoes and clothes are in rows.

Since the festive period of Christmas is approaching, the shop
expects to double the number of each item to sell. Express the
resulting matrix D.

1.2.3. Multiplication of matrices

Learning Activity 1.2.3

Two friends Agnes(A) and Betty(B) can buy sugar, rice and beans at one

1 3 2
or two supermarkets S1 and S2. Matrix M =   shows the number
 2 4 3
of kilograms of each of the three items(sugar, rice and beans)bought by

 1800 1600 
 
each of the two friends .Matrix P =  2500 2000  shows the price per
 1500 1200 
 
kilogram, in Rwandan francs, of sugar, rice and beans in supermarkets S1
and S 2 , respectively.

10 Mathematics | Student Book | Senior Five | Experimental Version

 a) Compare the number of columns of M to the number of rows of
P.
b) Calculate the shopping bill of each of the two friends at each of the
two supermarkets. Express the answer as matrix C .
c) How many rows and how many columns does C have?
d) Use matrix M and P to explain how each entry of C is obtained.

CONTENT SUMMARY
Let A and B be matrices of order n × p , and m × r , respectively. Matrices A and
B can be multiplied, in this order, if and only if p = m , that is, the number of
columns of the first matrix is equal to the number of rows of the second matrix.
In this case, we say that matrices A and B, in this order, are conformable for
multiplication. The product A × B is of order n × r , that is the product A × B
has the same number of rows as matrix A , and the same number of columns
as matrix B.

Practically, we proceed as follows:

1. Determine the order of the product:

A: n× p

B: p×r

A.B : n × r

2. Calculate the entries of the product as follow: Let A.B = ( cij ) . To determine
the entry cij , multiply the entries along the ith row of the first matrix by
the corresponding entries down the jth column of the second matrix and
add the products.
a a12   b11 b12 
In particular, if A =  11  and B =   , then
 a21 a22   b21 b22 

 a11 a12   b11 b12   a11b11 + a12b21 a11b12 + a12b22 

=A.B =  .   
 a21 a22   b21 b22   a21b11 + a22b21 a21b12 + a22b22 

Mathematics | Student Book | Senior Five | Experimental Version 11

Example 1.2.3.

 3 1
 2 3 0  
1. Consider matrices A =   and B =  −2 4 
 −1 4 1   2 −1
 

Determine whether A and B , in any order, are conformable for

multiplication or not.
b) In case, they are conformable for multiplication, find the order of the
products A.B and B. A .What do you conclude about the multiplication
of matrices?
c) Find the matrix A.B

Solution:
a)

A: 2×3

B: 3× 2

A.B : 2 × 2
A and B are conformable for multiplication, since the number of columns of A
is equal to the number of rows of B
In the same way,

B : 3× 2

A: 2×3

B.A : 3 × 3
B and A are conformable for multiplication, since the number of columns of B
is equal to the number of rows of A.
b) The order of the product A.B is 2 × 2 , and the order of the product
B. A is 3 × 3 .
Multiplication of matrices is not commutative. In general, for matrices A and
B, A.B ≠ B. A

12 Mathematics | Student Book | Senior Five | Experimental Version

  3 1
 2 3 0    2(3) + 3(−2) + 0(2) 2(1) + 3(4) + 0(−1)   0 14 
A.B 
c)=  .  −2 =
4  =   
 −1 4 1   2 −1  −1(3) + 4(−2) + 1(2) −1(1) + 4(4) + 1(−1)   −9 14 
 

5
 1 3 −1  
2. Consider matrices A =   , B =  2  and=
C (2 −3) .
 −2 5 3   4
 

a) Obtain the products ( A.B).C and A.( B.C )

b) Which property do you predict about multiplication of matrices?

Solution:

5
 1 3 −1    7 
=a) A.B =  . 2   
 −2 5 3   4  12 
 

7  14 −21 
.B).C   . ( 2=
( A= −3)  
12   24 −36 

5 10 −15 

   
.C  2  . ( 2 −=
B= 3)  4 −6 
 4  8 −12 
   

10 −15 
 1 3 −1    14 −21 
A.(B.C ) 
= =  .  4 −6   
 −2 5 3   8 −12   24 −36 
 

We can predict that multiplication of matrices is associative, that is, for all
matrices A, B and C , conformable for multiplication, ( A.B).C = A.( B.C )

Mathematics | Student Book | Senior Five | Experimental Version 13

  1 1  −1 1 
3. Let A =   ,and B =  .
 1 1  1 −1

a) Find the product A.B

b) What do you conclude?

Solution:

1 1  −1 1   0 0 
= a) A.B =  .   
1 1  1 −1  0 0 
A matrix, in which all the entries are zeros is said to be the null matrix or the
zero matrix.
For matrices, the equality A.B = 0 does not imply A = 0 or B = 0 , that is , the
product of two matrices can be the null matrix, yet none of the factors is a null
matrix.

Application activity 1.2.3

1. Determine whether matrices A and B, in this order, are conformable
for multiplication or not. In case, they are conformable, find the
product:

a) A = ( 3 1 12 ) and B = ( 4 2 1)

 3 2
b) A = ( 5 2 ) and B =  
7 6
2. A company’s input requirement over the next two months for two
inputs X and Y are given (in numbers of units of each input) by the

 2 1
matrix M =  .
 6 8
The company can buy these inputs from two suppliers, whose prices for
 4 2
the two inputs are given by the matrix N =   , where the two rows
5 1
represent the suppliers and the columns represent the prices.
Obtain the matrix for the total input bill for the next two months for both
suppliers.

14 Mathematics | Student Book | Senior Five | Experimental Version

 1.2.4. Inversion of matrices

Learning Activity 1.2.4

1 0 2  3 2 2 
  1 
=
Consider matrices A  0 1 −3= and B 5  3 2 −3 
 1 −1 0   1 −1 −1 
   
a) Find the product A.B , and write down its order.
b) Describe the entries of the product.
c) If such a matrix is called identity matrix, which of the following

 1 1 1 0 0 1
are identity matrices: i)   ii) (1) iii)   iv)  
 1 1 0 1 1 0

1 0
 
v)  0 1 
1 0
 
d) Matrix B is said to be the inverse of matrix A. When do we say that
matrix X is inverse of matrix M?

CONTENT SUMMARY
– A square matrix with each element along the main diagonal (from
the top left to the bottom right) being equal to 1 and with all other
elements being 0 is said to be the identity matrix, it is denoted by I;
For any square matrix A of order n × n , and the identity matrix I of order n × n
, we have:
A.I = A and I .A = A , that is, I is the identity element for multiplication of
matrices.

Mathematics | Student Book | Senior Five | Experimental Version 15

In particular,

Order Matrix A Identity I Product A= .I I= .A A

1× 1 (a) (1) (=
a ) . (1) (1=
) .( a ) ( a )
2× 2
a c  1 0  a c  1 0 1 0  a c  a c 
    =  .  =  .   
b d  0 1 b d  0 1 0 1 b d  b d 
3× 3  a a ' a ''   1 0 0   a a ' a '' 
    
 a a ' a ''  1 0 0  b b ' b ''  .  0 1 0  =  b b ' b '' 
     c c ' c ''   0 0 1   c c ' c '' 
   
 b b ' b ''  0 1 0
 c c ' c ''  0 0 1
    and

 1 0 0   a a ' a ''   a a ' a '' 

    
 0 1 0  .  b b ' b ''  =  b b ' b '' 
 0 0 1   c c ' c ''   c c ' c '' 
    

– If for a square matrix A of order n × n , there exists a square matrix B of

order n × n , such that A.B = I and B. A = I , where I is the identity matrix of
order n × n , then B is said to be the inverse of matrix A , and written B = A−1
– To find the inverse of a square matrix A ,of order n × n , by Gaussian method,
we, practically, proceed as follows:
Write ( A / I ) , a matrix of order n × (2n) , since the number of columns doubled,
but the number of rows is unchanged. Matrix ( A / I ) is an augmented matrix;
Transform the matrix ( A / I ) , using elementary row operations, to ( I / B ) .
Then B = A−1 .
– The following are the elementary row operations:
1. Interchanging two rows. For example, if row 1 and row 2 are interchanged,
then the entries of row 1 become the respective entries of row2, and vice
versa; we write R1 ↔ R2

2. Multiplying each entry of a non-zero real number k .For example, if the

entries of row3 are multiplied by, say 2, we write R3 → 2 R3
3. Adding to each entry of a row any multiple from any other row, for
example, R1 → R1 + kR2

16 Mathematics | Student Book | Senior Five | Experimental Version

If matrices B exists, then we say that A is invertible or regular;
If B does not exist, then we say that A is a singular matrix.

Example 1.2.4.
1 2 
Use elementary row operations to find the inverse of matrix A =   , and
1 −3 
verify that the product of the matrix A and its inverse is the identity matrix.

Solution:
1 2 1 0 
Consider the augmented matrix  
1 −3 0 1 

1 2 1 0  1 2 1 0
  R2 → R2 − R1  ;
1 −3 0 1   0 −5 −1 1 

 3 2
1 2 1 0 2  1 0
  R1 → R1 + R2  5 5;
 0 −5 −1 1  3  
 0 −5 −1 1 

 3 2 
 3 2 1 0 5 5 
1 0 5 5  R → − 1 R  ;
  2 5
2
0 1 1 − 1 
 0 −5 −1 1   
 5 5

3 2 
 5 5  1 3 2 
=
Therefore, the inverse matrix is A−1 =   
 1 − 1  5  1 −1
 
5 5
1 2  1  3 2  1 1 2   3 2 
The=product is A. A−1 = .     . 
1 −3  5  1 −1 5 1 −3   1 −3 
1 5 0 1 0
= =    
5  0 5   0 1 

Mathematics | Student Book | Senior Five | Experimental Version 17

 Application activity 1.2.4

1 1 1 
 
Given the matrix M = 1 2 3  ,
1 3 2 
 
a) Write down the identity matrix I of order 3 × 3 and determine
the augmented matrix ( M / I )

b) Use elementary row operations to transform ( M / I ) to ( I/ N )

c) Write down matrix M −1
d) Verify that M .M −1 = I and M −1.M = I

1.3. Determinants of square matrices

1.3.1.Definition and calculation of determinants of matrices
of orders 2 × 2 and 3 × 3

Learning Activity 1.3.1

1. Analyze the nature of entries in the rows (or columns) of the

 4 1  3 2
following square matrices A =   and B =  :
 5 3 12 8 
a) In matrix B, how are the entries in row 2 obtained from the
corresponding entries in row 1?
b) Can you suggest a means for characterizing a square matrix as
singular or not?

4 3 7
 
2. Determine whether matrix A =  1 6 7  is singular or not.
 3 1 4
 

18 Mathematics | Student Book | Senior Five | Experimental Version

CONTENT SUMMARY
a c 
A= 
Let  b d  be a square matrix of order 2 × 2 .Then the determinant of A

a c
is the unique real number denoted and defined by det =
A = ad − bc .
b d

 a a ' a '' 
 
If A =  b b ' b ''  is a square matrix of order 3 × 3 , then the determinant of A
 c c ' c '' 
 
a a ' a ''
is the unique real number denoted and defined by det A = b b ' b ''
c c ' c ''

= (ab ' c ''+ a ' b '' c + a '' bc ') − (cb ' a ''+ c ' b '' a + c '' ba ') : a sum of six terms, each term
having three factors
The following techniques can be used in evaluating the determinant of a

 a a ' a '' 
 
square matrix A =  b b ' b ''  of order 3 × 3 :
 c c ' c '' 
 

Sarrus’ method
1. Copy the first two columns as the fourth and fifth columns:

a a ' a '' a a '

b b ' b '' b b '
c c ' c '' c c '

2. Multiply the entries along the descending arrows and add the products
to obtain P1

Mathematics | Student Book | Senior Five | Experimental Version 19

P1 = ab ' c ''+ a'b''c'+ a''bc'
Multiply the entries along the ascending arrows and add the products to
obtain P2

P2 =cb ' a ''+ c ' b '' a + c '' ba '

The value of the determinant is P1 − P2 , that is,

a a ' a ''
det A =
b b ' b '' = (ab ' c ''+ a ' b '' c + a '' bc ') − (cb ' a ''+ c ' b '' a + c '' ba ')
P1 − P2 =
c c ' c ''

Expansion by cofactors
We have:

a11 a12 a13

a21 a22 a23 = (a11 a22 a33 + a12 a23 a31 + a13 a 21a 32 ) − (a31 a22 a13 + a32 a23 a11 + a33 a21 a12 )
a31 a32 a33

= a11 (a22 a33 − a32 a23 ) − a12 (a21a33 − a31a23 ) + a13 (a21a32 − a31a22 )

a22 a23 a a23 a a22

= a11 − a12 21 + a13 21
a32 a33 a31 a33 a31 a32

a22 a23 a a23 a a22

= a11 (−1)1+1 + a12 (−1)1+ 2 21 + a13 (−1)1+3 21
a32 a33 a31 a33 a31 a32

The determinant of the matrix remaining after deleting the row and the column
of an entry is called the minor of that element. Thus, if M ij is the matrix
remaining after deleting the i th row and the j th column of a square matrix

20 Mathematics | Student Book | Senior Five | Experimental Version

A = ( aij ) , then the minor of aij is det M ij

The cofactor of an entry aij is the number (−1)i + j det M ij .

Therefore, the determinant of a square matrix equals the sum of the products of
the entries on a row (or column) by their corresponding cofactors.
If the determinant of a square matrix is zero, then the matrix is singular; it has
no inverse.
If the determinant of a square matrix is not zero, then the matrix is invertible
or regular.

Example 1.3.1.

3 5 1
 
1. Calculate the determinant of the matrix A =  1 2 4  by expanding along:
 4 3 2
a) The first row  
b) The second column

Solution:

3 5 1
2 4 1 4 1 2
a) 1 2 4 = 3 −5 +1 = 3(−8) − 5(−14) + (−5) = 41
3 2 4 2 4 3
4 3 2

3 5 1
1 4 3 1 3 1
b) 1 2 4 =−5 +2 −3 =−5(−14) + 2(2) − 3(11) =41
4 2 4 2 1 4
4 3 2

2. Expanding along which row or column will make the calculation of the
determinant

7 23 1
0 −7 0 easier. Write down the value of the determinant.
13 −15 2

Mathematics | Student Book | Senior Five | Experimental Version 21

Solution:
The expansion along the second row will make the calculation of the
determinant easier.
7 23 1
7 1
In fact, 0 −7 0 = −7 −7(14 − 13) =
= −7
13 2
13 −15 2

Application activity 1.3.1

a c  a c
1. Differentiate between   and
b d  b d

1 1 1 
 
2. Calculate the determinant of the matrix M = 1 2 3  by:
1 3 2 
 

Sarrus’ method, (b) By expansion by cofactors

3. State whether the matrix X is singular or not if:

 5 0  4k −20k 
a) X =   b) X =   , where k is a constant real
 −3 9   −k 5k 
number.

22 Mathematics | Student Book | Senior Five | Experimental Version

 1.3.2. Properties of determinants

Learning Activity 1.3.2

1. Without calculation, predict the value of the determinant of each of
the following matrices:

 
 12 0 351
 1 3  
a) A =   b) B =  3 0 16 
 2 6  5 
− 0 −9 
 7 
 −1 2   3 −5 
2. Consider the matrices A =   and B =   .;
 3 −5   −1 2 

a) How are rows 1 and 2 of matrix B obtained from the rows of

matrix A ?

−1 2 3 −5
b) Calculate and
3 −5 −1 2

c) What do you conclude?

CONTENT SUMMARY
A square matrix can be changed into simpler for before calculating its
determinant through properties including the following:
m
1. If all the entries of a row or column of a square matrix are zeros, then the
determinant of the matrix is zero.
2. If all the entries of a row (or column) of a square matrix are multiplied
by a real number k , then the determinant of the matrix is multiplied by
a a ' a '' a a ' a ''
kb kb ' kb '' = k b b ' b ''
k .Thus, c c ' c '' c c ' c ''

Mathematics | Student Book | Senior Five | Experimental Version 23

 3. If two rows or columns of a square matrix are identical or proportional,
then the determinant of the matrix is zero.
4. If square matrix B is obtained by interchanging two rows or two columns
of square matrix A , then the determinant of B is the opposite of the
determinant of A .
5. 5)If a row or column of a square matrix B is obtained by adding or
subtracting any nonzero multiple of another row or column of matrix
A , the other rows or columns of B being the same as those of A ,then
the determinant of matrices A and B remains unchanged. Thus,

a a ' a '' a + ka ' a ' a ''

b b ' b ''= b + kb ' b ' b ''
c c ' c '' c + kc ' c ' c ''

Example 1.3.2.
 3 5 4
 
1. Calculate the determinant of the matrix A =  1 4 2  , then, wi hout
further calculation, find:  2 3 2
  t
3 4 5
1 2 4
2 2 3
a)
8 5 4
5 4 2
5 3 2
b)

Solution:
3 5 4
4 2 1 2 1 4
1 4 2 =3 −5 +4 =3(8 − 6) − 5(2 − 4) + 4(3 − 8) =−4
3 2 2 2 2 3
2 3 2

24 Mathematics | Student Book | Senior Five | Experimental Version

 a) Column 2 and column 3 of matrix A are interchanged ( C2 ↔ C3 )

3 4 5
Therefore, 1 2 4 = −(−4) = 4
2 2 3

b) Column2 is added to column1( C1 → C1 + C2 )

The determinant remains unchanged.

8 5 4 3 5 4
5 4 2 = 1 4 2 = −4
5 3 2 2 3 2

2. Evaluate

26 31 −1
53 9 42
0 0 0

Solution:

26 31 −1
53 9 42 =0, since row 3 consists of zeros only.
0 0 0

Mathematics | Student Book | Senior Five | Experimental Version 25

 Application activity 1.3.2
1. Check if the following statements are True or false:

a c ka kc
a) k =
b d kb kd

 a c   ka kc 
b) k  = 
 b d   kb kd 

1 1 1
2. Apply the indicated transformations to evaluate 1 2 1
1 1 2

a) R2 → R2 − R1 and R3 → R3 − R1
b) Write down the value of the determinant

1.4. Finding the inverse and solving simultaneous linear

equations
1.4.1. Inverse of a matrix

Learning Activity 1.4.1

1 1 1 
 
Given the matrix A = 1 2 1 
1 1 2 
 
a) Use augmented matrix and the following elementary row operations

i) R2 → R2 − R1 and R3 → R3 − R1

ii) R1 → R1 − R3

iii) R1 → R1 − R2

b) Write down A−1

26 Mathematics | Student Book | Senior Five | Experimental Version

 c) Perform the following:
i) Find det A
ii) Obtain matrix C ,where each entry of A is replaced by its cofactor
iii) Obtain matrix (denote it Adj ( A) ) by writing the entries of the
first row of C as respective entries of the first column of Adj ( A) ,
the entries on the second row of C as the respective entries of the
second column of Adj ( A) , and the entries on the third row of C as
the respective entries of the third column of Adj ( A)

1
iv) Write down matrix X = Adj ( A)
det A

v) Compare A−1 and X

CONTENT SUMMARY
Let A be a square matrix of order 2 × 2 or 3 × 3 .Then the inverse of A . Can also
be calculated through the following four steps:
1. Find the determinant of A , that is det A ;

2. Find the matrix C of cofactors of A : each entry of A is replaced by its

cofactor.

3. Find the adjoint of matrix A , denoted, Adj ( A) : the transpose of the

matrix of cofactors;

1
4. The inverse of matrix A is A−1 = Adj ( A) , where matrix A is
regular, or invertible. det A

The transpose of a matrix A of order n × p is the matrix denoted AT whose

rows are the columns of A and whose columns are the rows of A .

a c  −1 1  d −c 
In particular, if A =   is an invertible matrix, then A =  
b d  det A  −b a 
Example 1.4.1.
Find the inverse of each of the following matrices:

Mathematics | Student Book | Senior Five | Experimental Version 27

 3 5 1
 1 3  
A=  and B =  1 2 4 
 2 8  4 3 2
 

Solution:

1  8 −3 
A−1 =  
2  −2 1 

3 5 1
2 4 1 4 1 2
det B = 1 2 4 = 3 −5 + = 41
3 2 4 2 4 3
4 3 2

Matrix of cofactors:

 2 4 1 4 1 2
+ − + 
 3 2 4 2 4 3
 5  −8 14 −5 
1 3 1 3 5  
−
C= + − = −7 2 11 
 3 2 4 2 4 3   
   18 −11 1 
+ 5 1
−
3 1
+
3 5
 2 4 1 4 1 2 


 −8 −7 18 
 T 
Adjoint matrix: Adj (B)
= C=  14 2 −11
 −5 11 1 
 

 −8 −7 18 
1 
−1 
=
The inverse is B  14 2 −11
41  
 −5 11 1 

28 Mathematics | Student Book | Senior Five | Experimental Version

 Application activity 1.4.1
1. Find the inverse of matrices:
 20 6 
a) A =   and
 5 2

 5 0 2
 
b) M =  3 4 5 
 2 1 2
   3 7 2  2 −2 1 
 
2) Consider of the following matrices A =  1 0 2  and B =  3 1 6 
0 1 1  
   −1 1 1 
Find:  
i) A−1 and B −1

ii) (A )
−1 −1

(10A)
−1
iii)

1.4.2. Solving simultaneous linear equations using inverse of

a matrix

Learning Activity 1.4.2

A business makes floor tiles and wall tiles.
The table below shows the number of tiles of each type and the labor (in
hours) for making the tiles:

Material(tiles) Labor(hours)
Floor tiles 4 3
Wall tiles 11 1
Given that the total cost for floor tiles is 53 (thousand) FRW and the total cost
for wall tiles is 37(thousand) FRW, find the material cost and the labor cost
by answering the following questions:
a) Label x the material cost and y the labor cost, and then model the
problem by simultaneous linear equations in x and y

Mathematics | Student Book | Senior Five | Experimental Version 29

 b) Express the information in the table above as a matrix A of order
2 × 2 , the total floor tile cost and the total wall tile cost as a matrix B
of order 2 ×1 , and the material cost and the labor cost as a matrix X
of order 2 ×1
c) Perform the operation A. X = B and compare it to the simultaneous
equations obtained in part a)
d) Find the inverse matrix A−1 and the product A−1.B
e) Using X = A−1.B ,find the values of x and y .

CONTENT SUMMARY
The two simultaneous linear equations in two unknowns, x and y ,

 ax + by = c  a b  x   c 
 can be arranged, using matrices as,    =  
a ' x + b ' y =c'  a ' b '  y   c '

a b  x c
Let A =
=  , X   and B =   .
 a ' b '  y  c '
Then A. X = B .Multiplying both sides by A−1 , we have A−1 .( A. X ) = A−1 .B ,using
the associative property, the inverse property , and the identity property of
multiplication of matrices , we have:

( A−1. A). X = A−1.B

A−1.B
⇔ I .X =
A−1.B
⇔X=

In the same way, the three simultaneous linear equations in three unknowns,

 ax + by + cz = d

x , y and z  a ' x + b ' y + c ' z = d ' can be arranged, using matrices as,
a '' x + b '' y + c '' x = d ''

 a b c   x  d 
    
 a ' b ' c '  . y  =  d ' 
 a '' b '' c ''   z   d '' 
    

30 Mathematics | Student Book | Senior Five | Experimental Version

 a b c  x d 
     
=Let A =a ' b' c ' , X  y  and B =  d '  .Then A. X = B .Solving for X ,
 a '' b '' c ''  z  d '' 
     

X = A−1.B
Example 1.4.2.
Use matrices to solve the following simultaneous equations:

2 x − 3 y =
−8
a) 
 5x + y = −3

2 x − y + z =−1

b)  x + 3 y − z =
13
 x+ y+z = 3


Solution:
a) The system of the simultaneous equations can be expressed as

 2 −3   x   −8 
   =  
 5 1   y   −3 

 2 −3   x  −8 
Let A =
=  ; X   and B =   . The system becomes A. X = B ;
5 1   y  −3 

X = A−1.B ;

2 −3 1  1 3
det A= = 2(1) − 5(−3)= 17 ; A−1 =  ;
5 1 17  −5 2 

1  1 3   −8  1  −17   −1
=X  =   =     , that is x = −1 and y = 2
17  −5 2   −3  17  34   2 

Therefore, the solution set of the simultaneous equations is S= {(−1, 2)}

Mathematics | Student Book | Senior Five | Experimental Version 31

 b) The system of the simultaneous equations can be expressed as

 2 −1 1   x   −1
    
 1 3 −1  y  = 13 
1 1 1  z   3 
    

 2 −1 1   x  −1
     
A  1 3 −1 ; =
Let = X  y  and B =  13  .
1 1 1  z 3
     

The system becomes A. X = B ; X = A−1.B ;

2 −1 1
−1 1 2 1 2 −1
det
= A 1 3= −1 − + = 8 (Expansion along the third
row). 1 1 1 3 −1 1 −1 1 3

Matrix of cofactors:
 3 −1 1 −1 1 3 
 + − + 
 1 1 1 1 1 1 
 −1 1  4 −2 −2 
2 1 2 −1   
 −
C= + − = 2 1 −3 
 1 1 1 1 1 1  

   −2 3 7 
 −1 1 2 1 2 −1 
 + 3 −1 −
1 −1
+
1 3 


 4 2 −2 
 
The adjoint matrix is Adj ( A) =  −2 1 3  ,
 −2 −3 7 
 
 4 2 −2 
1 1 
A−=  −2 1 3 
8 
he inverse is  −2 −3 7  ;
T  4 2 −2   −1  16   2 
1   1    
X = −2 1 3   13  = 24  =
 3
8    8   
 −2 −3 7   3   −16   −2  , that is x = 2 , y = 3 and z = −2

32 Mathematics | Student Book | Senior Five | Experimental Version

 =
Therefore, the solution set of the simultan ous equations is
S {(2,3, −2)}
e

Application activity 1.4.2

Use matrices to solve the following simultaneous equations:

4 x + 7 y =
34
a) 
 3x − 2 y =
11

 x − 2 y + z =−2

b)  3 x + y − 2 z =7
 x + 3y − z = 2


1.4.3. Solving simultaneous linear equations using Cramer’s

rule

Learning Activity 1.4.3

 ax + by = c(1)
Consider the simultaneous linear equations 
a ' x + b ' y =c '(2)

1. a) Multiply both sides of equation (1) by b ' to get equation(3), and

multiply both sides of equation (2) by −b to get equation (4)
b) Perform the addition (3) + (4) to obtain equation (5)
c) Make x the subject of formula in equation (5) , precising the
condition for this operation to be valid(possible).Label (6) this
equation.
d) Express the numerator and the denominator of (6) as determinants
of matrices of order 2 × 2
2. a) Multiply both sides of equation (1) by −a ' to get equation (3') and
multiply both sides of equation (2) by a to get equation (4 ')
b) Perform the addition (3') + (4 ') to obtain equation (5')

Mathematics | Student Book | Senior Five | Experimental Version 33

 c) Make x the subject of formula in equation (5') , precising the
condition for this operation to be valid(possible).Label (6 ') this
equation.
d) Express the numerator and the denominator of (6 ') as
determinants of matrices of order 2 × 2
3. Use the formulas you have obtained above to solve the simultaneous
equations:
 3x + 2 y =4

5 x − 4 y =
14

CONTENT SUMMARY
To solve the two simultaneous linear equations in two unknowns x and y ,
Cramer’s rule requires to go through the following steps:

ax + by = c
1. Arrange the equations to get  and calculate the principal
a′x + b′y =c′
a b
determinant = D = ab′ − a′b
a ′ b′
If D = 0 , then the system has no solution or infinitely many solutions; the
system is not a Cramer’s system.

If D ≠ 0 , then the system is a Cramer’s system and has unique solution,

proceed to the next step:

c b
2. Write down and calculate: D=
x = cb′ − c′b and
c′ b′
a c
D=
y = ac′ − a′c
a ′ c′

Dx Dy
3. Write down and calculate x = and y = ; the solution set of the
D D
 D Dy  
simultaneous equations is S =  x , 
 D D  

34 Mathematics | Student Book | Senior Five | Experimental Version

 In the same way, for the three simultaneous linear equations in three

ax + by + cz = d

unknowns, x , y and z , a′x + b′y + c′z = d′
a′′x + b′′y + c′′x = d ′′


a b c
The principal determinant is D = a′ b′ c′
a′′ b′′ c′′

If D = 0 , then the system is not a Cramer’s system, it may have zero solution
or infinitely many solutions.

If D ≠ 0 , then the system is a Cramer’s system and has unique

 D Dy Dz  
solution; the solution set is S =  x , ,   , where
 D D D  

d b c a d c a b d
=Dx d= ′ b′ c′ , Dy a=′ d ′ c′ , Dz a′ b′ d ′
d ′′ b′′ c′′ a′′ d ′′ c′′ a′′ b′′ d ′′

Example 1.4.3.

3 x + 2 y − 2 z = 4

Use Cramer’s rule to solve the simultaneous equations:  x + 3 y + z = 7
2 x + y − z = 11


3 2 −2
3 1 1 1 1 3
Solution: D = 1 3 1 = 3 −2 −2 = 4
1 −1 2 −1 2 1
2 1 −1

Mathematics | Student Book | Senior Five | Experimental Version 35

 4 2 −2
3 1 7 1 7 3
Dx = 7 3 1 = 4 −2 −2 = 72
1 −1 11 −1 11 1
11 1 −1
3 4 −2
7 1 4 −2 4 −2
1 7 1 =
Dy = 3 − +2 −36
=
11 −1 11 −1 7 1
2 11 −1
3 2 4
3 7 2 4 2 4
Dz = 1 3 7 = 3 − +2 = 64
1 11 1 11 3 7
2 1 11

Dx 72
x
= = = 18
D 4
Dy −36
y= = = −9
D 4
Dz 64
z =
= = 16
D 4

The solution set is=S {(18, −9,16 )}

Application activity 1.4.3
Use Cramer’s rule to solve the following simultaneous equations:

4 x + 7 y =
34
a) 
 3x − 2 y =
11

 x − 2 y + z =−2

b)  3 x + y − 2 z =7
 x + 3y − z = 2


36 Mathematics | Student Book | Senior Five | Experimental Version

 End of unit assessment 1

1. Write down the order of each of the following matrices:

 −3 4 2 
a) A =   B
b)= (4 −1)
 1 5 −1

2. Given that matrices A and B are equal, find the values of the letters:

 3m + 2 2   m − 3 2
a) A =   and B =  
 2k + 1 1   5 6

 x −1 −3   6x + 5 2 
b) A =   and B =  
 x + 4z 3y + 4   1 1

3. Perform each of the following operations:

 −2 3   1 4 
a) 3   − 2 
 1 2   −1 −1

1 3
 −1 3 1   
b)    1 −1
 2 4 −1  −1 2 
 

4. Invertible 2 × 2 matrices A, B and X are such that 4 A − 5 BX =

B
a) Make X the subject of the formula
b) Find X if A = 2 B

2 0 1 1 0 1 
   
5. a) Given that A =  3 0 0  , and B = 1 2 1  , find the products
5 1 1 1 1 0 
   
A.B and B. A

Mathematics | Student Book | Senior Five | Experimental Version 37

  1 −1 1   1 2 3 
  
c) Calculate the product  −3 2 −1 .  2 4 6 
 −2 1 0   0 2 3 
  
6. Find the inverse matrix of each of the following matrices:

2 0 1
 −1 4   
a) A =   b) M =  3 0 0 
 2 3 5 1 1
 

1 2 x2   1 4 1
  T  
7. a) The transpose of matrix M =  4 1 0  is M =  2 1 1  .
1 x +3 8   4 0 8
   

Find the value of x

a c  a' c'
b) Use A =   and B =   to derive the formulas for :
b d   b ' d '
i) ( AT )T : the transpose of the transpose of a matrix

ii) ) ( A.B)T ; the transpose of a product

8. Evaluate the following:

3 2 1
12 6
a) a) b) 0 2 −5
5 4
−2 1 4

2 0 0
9. a) In the calculation of 1 2 0 , along which row or column is it
3 5 6

advantageous to expand by cofactors? Write down the value of

the determinant

38 Mathematics | Student Book | Senior Five | Experimental Version

 2 1 2
b) i) Evaluate 1 2 0 , using expansion by cofactors along row 2.
3 5 6

2 1 2
ii) Name the transformation applied on the determinant 1 2 0 to
3 5 6
2 1 7
obtain the determinant 1 2 4 .
3 5 17

2 1 7
iii) write down the value of 1 2 4
10. 3 5 17
a) Use inverse matrix to solve the simultaneous linear equations:

 x+ y−z = 0

 x + 2 y + 3z =
14
2 x + y + 4 z =
16

b) Use Cramer’s rule to solve the simultaneous equations:

 x+ y+z = 6

 2x + y − z = 1
3 x + 2 y + z =
10


Mathematics | Student Book | Senior Five | Experimental Version 39

 Unit 2 Differentiation/Derivatives

Key Unit competence: Solve Economical, Production, and Financial related

problems using derivatives.

Introductory activity

y 3 x − 2 (1), and =
Consider functions = y x 2 + 1 (2)
a) complete the following table for each of the two functions:

x1 x2 ∆x = x2 − x1 y1 y2 ∆y = y2 − y1 ∆y
∆x

1 2
1 1.5

1 1.1
… … … … … … …

∆y
b) How is the quantity called?
∆x
∆y
c) Compare for function (1) and for function (2)
∆x
∆y
d) Predict a formula for lim
∆x →0 ∆x

40 Mathematics | Student Book | Senior Five | Experimental Version

2.1 Differentiation from first principles
2.1.1. Average rate of change of a function

Learning Activity 2.1.1

Suppose that the profit by selling x units of an item is modeled by the

equation, P( x) = 4 x 2 − 5 x + 3 , and x assumes values 2 and 5 ,respectively.
Find:
a) The change in x
b) The values of P for x = 2 and x = 5 , respectively. Hence, find the
change in P
c) Find the ratio of the change in P to the change in x
d) Give a word with the same meaning as ratio

CONTENT SUMMARY

The average rate of change of function , y = f ( x) as the independent variable

∆y f ( x2 ) − f ( x1 )
x assumes values from x1 to x2 , is the quantity defined by =
∆x x2 − x1
Example 2.1.1.
The cost (in thousand FRW) of producing a certain commodity is modeled by
the function C ( x=
) 40 + x . Find the average rate of change of the cost when
the production level changes from x = 64 to x = 100 .

Solution:

The change in the independent variable is ∆x= 100 − 64= 36 ;

C (100) =
40 + 100 =
50 and C (64) =40 + 64 =48 ; The change in the
dependent variable is ∆C = C (100) − C (64) = 50 − 48 = 2 , The average rate of

∆C 2 1
change in the cost is = =
∆x 36 18

Mathematics | Student Book | Senior Five | Experimental Version 41

 Application activity 2.1.1

1. In a Forex bureau, for 100 units of currency y , you obtain 107,500

units of currency x . Assuming that the two currencies are related
by a linear equation, find the average rate of change of currency y ,
with respect to currency x .

1
2. ) x 2 − as x
Find the average rate of change of function f ( x=
x
assumes values from 2 to 4 .

2.1.2. Instantaneous rate of change of a function

Learning Activity 2.1.2

) x 2 + 1 , and the changes in x from x0 = 2 to

y f ( x=
Consider function =
x1 , where x1 assumes consecutively values
= x1 2.1;
= x1 2.01;
= x1 2.001;...
a) Complete the following table:

x1 2.1 2.01 2.001 etc x1 → ...

∆x = x1 − x0 etc ∆x → ...

∆y = y1 − y0 etc /////////////

∆y etc ∆y
→ ...
∆x ∆x

∆y
→b
b) How is the statement “If ∆x → a then ∆x ” written in terms
of limits?

∆y
c) Find, in terms of x , lim for y = 3 x 2 at any value of x .
∆x →0 ∆x

42 Mathematics | Student Book | Senior Five | Experimental Version

CONTENT SUMMARY

The instantaneous rate of change of function y = f ( x) at x = x0 is the value

∆y
lim
f ∆x →0 ∆x
or x = x0 .
f
o
By finding the instantaneous rate of change of function y = f ( x) by using limit,
we say that we are differentiating function y = f ( x) from first principles.
Equivalently, the instantaneous rate of change of function y = f ( x) at x0 is the

f ( x0 + ∆x) − f ( x0 )
number, lim , provided it exists.
∆x →0 ∆x
This number is called the derivative of function y = f ( x) at x0 ; it is denoted by
f '( xo ) . More generally, the derivative of function y = f ( x) is denoted by f '( x)

dy
or y ' or
dx
Example 2.1.2.

1
=
Differentiate function ( x)
y f= from first principles.
x
Solution:

f ( x + ∆x) − f ( x)
f '( x) = lim
∆x →0 ∆x

1 1
−
= lim x + ∆x x
∆x →0 ∆x

−1
= lim
∆x →0 x ( x + ∆x )

−1
=
x2

Mathematics | Student Book | Senior Five | Experimental Version 43

 Application activity 2.1.2
1. Find the instantaneous rate of change of the total cost
C (Q) = Q 2 + 7Q + 23 ,with respect to the number Q of units sold

2. Differentiate, from first principles, f (=

x) x −3

2.2. Rules for differentiation

2.2.1. Differentiation of polynomial functions

Learning Activity 2.2.1

Consider the following functions:

1 4
f ( x) = 3x 2 − 2 x + x + 1 ; g ( x) = x + 5 x − 7 and h( x)= 4 x −3 + 7 x − 5
3
a) Which of the three functions is a polynomial function?
b) Describe the quantities and the operations involved in a
polynomial
c) Obtain, from first principles, the derivative of each of the following:
i) A constant function C

ii) A sum of two functions u ( x) and v( x)

iii) A product of a real number k by a function u ( x)

n
iv) x; x ; x , and then predict the derivative of x , where x is the variable
2 3

and n a positive integer.

d) Write down the derivative of P( x) = a0 + a1 x + a2 x 2 + ... + an x n

, where a0 , a1 , a2 ,..., an are constant real numbers, 0,1, 2,..., n are
positive integers, and x is the variable.

44 Mathematics | Student Book | Senior Five | Experimental Version

CONTENT SUMMARY
A polynomial function in one variable x is a function of the type
P( x) = a0 + a1 x + a2 x 2 + ... + an x n .Thus, a polynomial consists of a variable raised

to different positive integral powers, powers are multiplied by constants and

the products are added, or subtracted. It can be shown, from first principles,
that:
d
1. (C ) = 0 : the derivative of a constant is 0 ;
dx
d d d
2. (u + =
v) (u ) + (v) : the derivative of a sum equals the sum of
dx dx dx
derivatives of the terms, where u and v are functions of variable x ;
d d
3. (ku ) = k (u ) , where k is a constant and u is a function of variable
dx dx
x : the derivative of the product of a constant real number by a function
equals the product of the real number by the derivative of the function;
d n
4. ( x ) = nx n −1 ; the derivative of the nth power of the variable equals the
dx
pr
oduct of the exponent by the (n-1)th power of the variable. From th
properties above, it follows that: e

d
(a0 + a1 x + a2 x 2 + ... + an x n ) = a1 + 2a2 x + ... + nan x n −1
dx

Example 2.2.1.
Find the derivative of each of the following polynomials:
3 2
a) f ( x) = 2 x − 5 x + 11x − 3
1 3
b) g ( x) = − x 3 + x 2 − 25
3 4
Solution:
2
a) f '(x) = 6 x − 10 x + 11
b) g '( x) = 3
− x2 + x
2

Mathematics | Student Book | Senior Five | Experimental Version 45

 Application activity 2.2.1

dz dP
Find the derivative or of:
dt dQ
3 5
a) z =2 − 8t + 3t
b) P = 5Q 4 + 3Q − 7

2.2.2. Differentiation of product functions

Learning Activity 2.2.2

Let u =−3 x 2 + 2 and =

v 4x − 9 :
a) i) Expand the product uv

d
ii) Calculate (uv)
dx
du dv
b) i) Find and
dx dx
du dv
ii) Calculate: v +u
dx dx
d du dv
c) Compare (uv) and v +u
dx dx dx

CONTENT SUMMARY
To find the derivative of the product of two functions u and v of independent
variable x , one can proceed as follows: Either expand the product, and then
find the derivative of the expansion, Or, calculate the derivative of each factor,

du dv
and then obtain the products v and u , and finally, consider the sum
dx dx
du dv d d d
v + u , that is, = (uv) v (u ) + u (v)
dx dx dx dx dx
It is the matter of choosing the more convenient method.

46 Mathematics | Student Book | Senior Five | Experimental Version

Example 2.2.2.
Find the derivative of each of the following products:
(2 x 3 − 5 x 2 )(11x − 3)
a) f ( x) =

1 3
− x 3 ( x 2 − 25)
b) g ( x) =
3 4
Solution:

d d
u 2 x 3 − 5 x 2 and=
Let= v 11x − 3 . Then, u ) 6 x 2 − 10 x and
(= (v) = 11
dx dx
Substituting these values in the product rule formula,

d
(uv) = (11x − 3)(6 x 2 − 10 x) + 11(2 x3 − 5 x 2 )
dx
=88 x3 − 183 x 2 + 30 x

1 3 2 d d 3
Let u = − x 3 and
= v x − 25 , Then (u ) = − x 2 and (v ) = x
3 4 dx dx 2
Substituting these values in the product rule formula,

d 3 1 3
(uv
= ) ( x 2 − 25)(− x 2 ) + (− x3 )( x)
dx 4 3 2

5
− x 4 + 25 x 2
=
4

Application activity 2.2.2

dz dP
Find the derivative or of:
dt dQ

(2 8t 3 )(4 + 3t 5 )
a) z =−
b) P 5Q 4 (3Q − 7)
=

Mathematics | Student Book | Senior Five | Experimental Version 47

 2.2.3. Differentiation of power functions

Learning Activity 2.2.3

y (2 x + 1)3
Let=

dy
a) Expand y and then calculate
dx
d
b) i)Find (2 x + 1)
dx
d
ii) Calculate 3(2 x + 1) 2(2 x + 1)
dx
iii) Describe how to obtain 3(2 x + 1) 2 from (2 x + 1)3

d n
c) Predict a rule for finding (u ) , where u is function of variable x
dx

CONTENT SUMMARY
n
Let y = u , where u is function of variable x and n a rational number. It can be
d n du
(u ) = nu n −1
shown that dx dx

To find the derivative of the nth power of function u of independent variable

x , one can proceed as follows: Either expand the power, and then find the
derivative of the expansion,
Or, calculate the derivative of the base, and then apply the formula

d n du
(u ) = nu n −1
dx dx
It is the matter of choosing the more convenient method.

Example 2.2.3
Find the derivative of each of the following powers:
a) f ( x)= (2 x3 + 3 x 2 − 5 x + 1)6

( x)
b) g= 3x + 2

48 Mathematics | Student Book | Senior Five | Experimental Version

Solution:
) 6(2 x3 + 3 x 2 − 5 x + 1)5 (6 x 2 + 6 x − 5)
a) f '( x=
1 1
d 1 −1 d
b) We have: 3 x + 2 = (3 x + 2) 2 ’ ( 3 x + 2)= (3 x + 2) 2 (3 x + 2)
dx 2 dx
1
1 −
= (3 x + 2) 2 (3)
2

3
=
2 3x + 2

Application activity 2.2.3

Find the derivative of:
y (7 x + 8) 2
a) =
y (4 x − 5)3
b) =

c) P 5Q 4 (3Q − 7)
=

2.2.4. Differentiation of the composite function (The chain

rule)

Learning Activity 2.2.4

y (2 x + 1)3
Let=

dy
a) Use the power rule to calculate
dx
b) i) Determine two functions u ( x) and v(u) such that y = vu ,
that is y = v[u ( x)]

du dv dy
ii) Find and =
dx du du
dy du
iii) Calculate .
du dx

Mathematics | Student Book | Senior Five | Experimental Version 49

 dy dy du
c) Compare and .
dx du dx
iii) Describe how to obtain 3(2 x + 1) 2 from (2 x + 1)3

d n
d) Predict a rule for finding (u ) , where u is function of variable
x dx

CONTENT SUMMARY
Consider the following diagram:

If function y = f ( x) can be expressed as y = v[u ( x)] , where u ( x) nd v(u ) are

dy dy du a
= .
functions to determine, then, it can be shown that: dx du dx . This formula is
known as the chain rule.

Example 2.2.4.
Use the chain rule to find the derivative of:
y (4 x + 5)6
a) =

b)=y 3x + 2

Solution:

du
u 4 x + 5 .Then
a) Let = = 4 , the function becomes y = u 6 . We have
dx
dy
u 5 6(4 x + 5)5 ,
= 6=
du

dy dy du
From the chain rule, = . =6(4 x + 5)5 (4) =24(4 x + 5)5
dx du dx

50 Mathematics | Student Book | Senior Five | Experimental Version

 1
du
u 3 x + 2 . Then,
b) Let = = 3 , the function becomes y = u 2 We have
dx
1
dy 1 − 2 1 1
= = u = ,
du 2 2 u 2 3x + 2

dy dy du 1 3
= =
From the chain rule, . = (3)
dx du dx 2 3 x + 2 2 3x + 2

Application activity 2.2.4

Use the chain rule to find the derivative of:
y (7 x + 8) 2
a) =
y (4 x − 5)3
b) =

2.2.5. Differentiation of quotient functions

Learning Activity 2.2.5

3x 2
Consider the function y =
2x +1
a) i) Express y in the form y = u.v −1 , stating the value of u and the
value of v

ii) Use the product rule to find the derivative of y = u.v −1 and, express

N ( x)
your answer with positive exponents, in the form of a fraction
D( x)
d d
(3 x + 1) (3 x 2 ) − (3 x 2 ) (3 x + 1)
b) Calculate dx dx
(3 x + 1) 2
du dv
v −u
d u dx dx
c) Compare ( ) and 2
dx v v
d u
d) Predict a rule for finding ( ) , where u and v are functions of
variable x dx v

Mathematics | Student Book | Senior Five | Experimental Version 51

CONTENT SUMMARY

It can be shown that, if u and v are functions of variable x , and v( x) ≠ 0 ,

du dv
v −u '
d u dx dx , that is,  u  u 'v − v 'u
Then  =   =
dx  v  v2 v v2

Example 2.2.5.
Use the quotient rule to find the derivative of:

x 2 − 3x + 2
a) y =
2x + 1

2− x
b) y =
x

Solution:

Let u = x 2 − 3 x + 2 and =
v 2x +1 .

du dv
Then = 2 x − 3 and = 2.
dx dx

dy (2 x + 1)(2 x − 3) − 2(x 2 − 3 x + 2)
We have =
dx (2 x + 1) 2

2 x 2 + 3x − 7
=
(2 x + 1) 2
Let u= 2 − x and v = x .

du dv 1
Then = −1 and = .
dx dx 2 x

2− x
− x−
=
We have:
dy
= 2 x −x − 2
dx x 2x x

52 Mathematics | Student Book | Senior Five | Experimental Version

 Application activity 2.2.4
Use the quotient rule to find the derivative of:

5x − 6
a) y =
7x − 4
3x 4
b) y =
2 + 5x

2.2.6. Differentiation of logarithmic functions

Learning Activity 2.2.6

Consider function f ( x) = ln x
a) Using a calculator, complete the following table:

x0
1 2 3 …
∆x
∆y ∆y ∆y
∆x ∆x ∆x
0.1 ln(1 + 0.1) − ln1 ln(2 + 0.1) − ln 2 ln(3 + 0.1) − ln 3
= = =
0.1 0.1 0.1
= = =
∆y ∆y ∆y
∆x ∆x ∆x
0.01 ln(1 + 0.01) − ln1 ln(2 + 0.01) − ln 2 ln(3 + 0.01) − ln 3
= = =
0.01 0.01 0.01
= = =
∆y ∆y ∆y
∆x ∆x ∆x
0.001 ln(1 + 0.001) − ln1 ln(2 + 0.001) − ln 2 ln(3 + 0.001) − ln 3
= = =
0.001 0.001 0.001
= = =
…

b) Use the table above to write the values of f '(1); f '(2); f '(3);... f '( x)

Mathematics | Student Book | Senior Five | Experimental Version 53

CONTENT SUMMARY

1
The derivative of function= ( x) ln x is
y f= f '( x) (ln
= = x) ' , that is
x
d 1
( ln x ) = , where x > 0 . More generally, if y = ln[u ( x)] , where u ( x) is function
dx x
d 1 du u'
of variable x , then from the chain rule, [ lnu(x) ] = . , that is [ ln u ( x) ] ' = .
dx u dx u
1 1 1
In the same way, if log a x = .ln x , then=( log a x ) ' = (ln x) ' , where
ln a ln a x ln a
a > 0 and a ≠ 1 , and x > 0 .
More generally,

u'
If y = log a u ( x) , where u ( x) is function of variable x , then [ log a u ( x) ] ' =
u ln a
Example 2.2.6.
Find the derivative of:
a)=y ln( x 2 + 3)
b)=y log(1 − x 2 )

Solution:

2x
a) y ' = 2
x +3
−2 x
b) y ' =
(1 − x 2 ) ln10

Application activity 2.2.6

Find the derivative of:
a) y = ln 4 x3
b) y log(5 x + 6)
=

54 Mathematics | Student Book | Senior Five | Experimental Version

 2.2.7. Differentiation of exponential functions

Learning Activity 2.2.7

1. Consider the equality y = a x , where a > 0 and a ≠ 1

a) Make x the subject of the formula
b) Differentiate, with respect to x ,both side of the equation obtained
in part (a)
c) Make y ' the subject of the formula in part (b)
d) Complete:

i) (a x ) ' = ...

ii) in particular, (e x ) ' = ...

2. Use the chain rule to obtain the formula for:

i) (a u ) '

ii) (eu ) ' , where u ( x) is function of variable x

CONTENT SUMMARY

d x
=
he derivative of function ( x) e x is f =
y f= x
'( x) (e= ) ' e x , that is dx
e = ex ( )
T u ( x)
. More generally, if y = e , where u ( x) is function of variable x , then from
d u ( x) du
the chain rule, e  = eu ( x ) , that is eu ( x )  ' = u ' eu . In the same way, if a x ,
dx dx
( )
then a x ' = a x ln a , where a > 0 and a ≠ 1 , and x > 0 . More generally, If y = a u ( x )

where u ( x) is function of variable x , then  a u ( x )  ' = u ' a u ln a

Example 2.2.7.
Find the derivative of:
2
+3
a) y = e x
2
b) y = 101− x

Mathematics | Student Book | Senior Five | Experimental Version 55

Solution:
2
+3
a) y ' = 2 xe x
2
b) y ' = (−2 x)(101− x ) ln10

2.3. Some applications of derivatives in Mathematics

2.3.1. Equation of the tangent to the graph of a function at a
point.

Learning Activity 2.3.1

Let y = f ( x) be a numerical function, and A ( x0 , f ( x0 ) ) , B ( x0 + ∆x, f ( x0 + ∆x) )

two points on the graph of y = f ( x) , see the diagram below:

a) Write down the gradient of the straight line through points A and B
b) As point B moves along the curve towards point A :
i) The change in x , that is ∆x approaches which value?
ii) The gradient of the straight line through points A and B approaches
which value? (Express your answer in terms of the derivative of
function f
iii) How becomes the position of the line through points A and B with
respect to the graph of function y = f ( x)
c) Write down the equation of the straight line through points A and T .

56 Mathematics | Student Book | Senior Five | Experimental Version

CONTENT SUMMARY

The derivative of function y = f ( x) at x0 is the gradient m of the geometric

tangent to the graph of y = f ( x) at point A, that is, if m is the gradient of the
tangent then, m = f '( x0 ) .

Therefore, the equation of the tangent to the graph of function y = f ( x) at x0

is:

y − f ( x0 )= f '( x0 )( x − x0 ) , since the tangent passes through point A( x0 , f ( x0 ))

Example 2.3.1.

Find the equation of the tangent to the graph of function f ( x) at x0 :

−2 x 2 + 80 x ; x0 = 10
a) f ( x) =
b) f ( x) =
200 − 40 ln( x + 1) ; x0 = 19

Solution:

a) f ( x0 ) = −2(10) 2 + 80(10) =
f (10) = 600 ; f '( x) =−4 x + 80 ;
f '(10) =
−4(10) + 80 =
40 ;
The equation of the tangent is: y − 600 = 40( x − 10) ; or, equivalently,=
y 40 x + 200
b) f ( x0 )= f (19)= 200 − 40 ln(19 + 1)= 200 − 40 ln 20 ≈ 80.17 ;

−40 −40
f '( x) = ; f '(19) = = −2 ;
x +1 19 + 1
The equation of the tangent is: y − 80.17 =
−2( x − 19) ; or, equivalently,
y= −2 x + 98.17

Application activity 2.3.1

Find the equation of the tangent to the graph of function f ( x) at x0 :

x
−
a) f ( x) 15
= = xe 3 ; x0 6 (leave your answer in terms of e )

1
b) f (x) = 10- x ; x0 (Leave your answer in terms of surds)
2

Mathematics | Student Book | Senior Five | Experimental Version 57

 2.3.2. Hospital’s rule.

Learning Activity 2.3.2

f ( x) f ( x0 ) 0
Let numerical functions f ( x) and g ( x) be such that lim = =
x → x0 g ( x ) g ( x0 ) 0
∞
or
∞
a) Determine the equations of the tangents to the graphs of functions
f ( x) and g ( x) at x0 , in the form =
y ax + b and =
y cx + d

f ( x)
b) Calculate lim , substituting f ( x) , and g ( x) , respectively by
x → x0 g ( x)
ax + b and cx + d , obtained from (a) and simplify.

f ( x)
c) Express lim in terms of a limit involving f '(x 0 ) and g '( x0 )
x → x0 g ( x)

CONTENT SUMMARY
f ( x) f ( x0 ) 0
lim = =
If num ric l functions f ( x) and g ( x) are such that g ( x) g ( x0 ) 0 or
x → x0

∞ e a
∞ , then to remove the indetermination, we proceed as follows, through

Hospital’s rule:
1. Differentiate separately, the numerator and the denominator, to get
f '(x) and g '( x) ;

f '( x) f '( x0 )
2. Calculate lim =
x → x0 g '( x) g '( x0 )

f ( x) f '( x0 )
3. Then lim =
x → x0 g ( x ) g '( x0 )
Note that: - the process can be repeated if necessary;

0 ∞
– Hospital’s rule is used only if we have indetermination or
0 ∞

58 Mathematics | Student Book | Senior Five | Experimental Version

Hospital’s rule, is not the quotient rule for differentiation, that is

'
f ( x)  f ( x) 
lim ≠ lim  
x → x0 g ( x ) x → x0 g ( x )
 

– Before applying Hospital’s rule, ensure that you have indetermination

0 ∞
or
0 ∞
Example 2.3.2.
Evaluate the following limits:

1 − ln x
lim
x →e x − e

e3 x − 1
lim
x →0 x2
Solution:

1 − ln x 1 − ln e 1 − 1 0
a) lim = = = : indeterminate case
x−e
x →e e−e e−e 0
Applying Hospital’s rule,

1
−
1 − ln x (1 − ln x) ' 1 1
We have: lim lim
= lim x =
= − lim = −
x →e x − e x → e ( x − e) ' x →e 1 x →e x e
e3 x − 3 x − 1 e3(0) − 3(0) − 1 1 − 1 0
b) lim = = = : indeterminate case
x →0 x2 02 02 0

Applying Hospital’s rule,

e3 x − 3 x − 1 (e3 x − 3 x − 1) ' 3e3 x − 3 0

We have: lim = lim = lim
= : Indeterminate
x →0 x2 x →0 ( x2 ) ' x →0 2x 0
case.

3e3 x − 3 (3e3 x − 3) ' 9e3 x 9

Repeating the process: lim = lim = lim =
x →0 2x x →0 (2 x) ' x →0 2 2

Mathematics | Student Book | Senior Five | Experimental Version 59

 Application activity 2.3.2
Evaluate the following limits:

x −1
a) lim
x →1 ( x − 1)e x

2 x−2
b) lim e −1
x →1 ln(5 x − 4)

End of unit assessment 2

1. For function f ( x) = ln x , find:

a) The average rate of change of f ( x) as x assumes values from 1

to 1.2
b) The instantaneous rate of change at x0 = 1
2. Find, from first principles, the derivative of:

a) f ( x) = 3x 2

−5
b) f ( x) =
x
3. Find the derivative of:

y 30 x − 0.5 x 2
a) =

1
b) y =+ 2 x − 3
x
4. Differentiate, with respect to t :
a) s 3t 4 (2t − 5)
=
b) s =(t 7 − 4)(t 5 + 11)

dy
5. Find ,using the chain rule, if:
dx
a) y= x2 − x + 2
b)=y (3 x 4 + 7)6

60 Mathematics | Student Book | Senior Five | Experimental Version

 dQ
6. Find if:
dP

5P 2 − 9 P + 8
a) Q =
P2 + 1
6P − 7
b) Q =
8P − 5
7. Differentiate, with respect to x :

(3 x + 4)3
a) y =
5x −1
2
 2x +1 
b) y =  
 3x − 5 
8. Find the derivative of;

a) y = 21−3 x

b) y = 3ln 2 (5 x )

9. Find the equation of the tangent to the graph of function f ( x) at x0

a) f ( x) = −2e3 x ; x0 = 0

x
b) f ( x) =; x0 = e 2
ln x
10. Calculate the following limits:

e 2 x − ln( x + e)
a) lim
x →0 x
ex
b) lim
x →0 e x + 1

Mathematics | Student Book | Senior Five | Experimental Version 61

 3
Applications of
Unit derivatives in Finance
and in Economics

Key Unit competence: Apply differentiation in solving Mathematical problems

that involve financial context such as marginal cost,
revenues and profits, elasticity of demand and supply

Introductory activity

A can company produces open cans, in cylindrical shape, each with

constant volume of 300 cm3 . The base of the can is made from a
material that costs 50 FRW per cm 2 , and the remaining part is made of
material that costs 20 FRW per cm 2 .
a) Express the height of one can as function of the base radius x
of the can.
b) Express the total cost of the material to make a can, as function
of the base radius x of the can.
c) Find the dimensions of the can that will minimize the total cost
of the material to make a can.

3.1. Marginal quantities

3.1.1. Marginal cost

Learning Activity 3.1.1

A company found that the total cost y of producing x items is given by

y = 3 x 2 + 7 x + 12 .
a) Find the instantaneous rate of change in the total cost, when x = 3
b) How is the instantaneous rate of change in the total cost called?

62 Mathematics | Student Book | Senior Five | Experimental Version

CONTENT SUMMARY
The marginal cost is the instantaneous rate of change of the cost.
It represents the change in the total cost for each additional unit of production.
Suppose a manufacturer produces and sells a product. Denote C(q) to be the
total cost for producing and marketing q units of the product. Thus, C is a
function of q and it is called the (total) cost function. The rate of change of C
with respect to q is called the marginal cost, that is,

dC
Marginal Cost =
dq
Example 3.1.1.
A radio manufacturer produces x sets per week at a total cost of

1 2
y
= x + 3 x + 100 FRW.
25
Find the marginal cost for x = 30

Solution:
d 1 2 2
( x + 3 x + 100)= x+3
dx 25 25

2
For, x = 30 , the value of the marginal cost is (30) + 3 =5.4 per unit.
25

Application activity3.1.1

1. The total cost C of producing x items of a commodity is

C = 4 x − x 2 + 2 x3 FRW. Find the marginal cost of the commodity.
2. The production function of a commodity is given by

1
Q = 40 P + 3P 2 − P 3 , where Q is the total output and P is the
3
number of units of input. Find the value of the marginal product for
P = 10

Mathematics | Student Book | Senior Five | Experimental Version 63

 3.1.2. Marginal revenue

Learning Activity 3.1.2

P 100 − Q .
A firm has the following demand function: =
Find: a) in terms of Q , the total revenue function
b) The instantaneous rate of change of the total revenue when Q = 11 .

CONTENT SUMMARY

If y = C ( x) is cost of producing x units of a product, then R( x) ,the total

revenue generated by selling x units of the product, is given by R ( x) = x.C ( x)
: the product of the number of units produced by the cost of producing the
units. Then, the marginal revenue is the instantaneous rate of change in the
dR
tota revenue, that is dx .
l
Example 3.1.2.

P 16 − Q ,find the marginal

Given the demand function,= total revenue for
Q=7

Solution:

(Q) Q 16 − Q ; The marginal revenue is

The total revenue is R=

dR −3Q + 32
= Q ' 16 − Q + Q( 16 − Q=
)' ,
dQ 2 16 − Q

dR −3(7) + 32 11
= 7,
For Q= =
dQ 2 16 − 7 6

Application activity 3.1.2

1. y 30 − 2 x , find the marginal revenue

Given the demand function =
2. Find the marginal revenue associated with the supply function
P = Q 2 + 2Q + 1 for Q = 10

64 Mathematics | Student Book | Senior Five | Experimental Version

3.2. Minimization and maximization of functions
3.2.1. Minimization of the total cost function

Learning Activity 3.2.1

Consider the following problem: A can company produces open cans, in
cylindrical shape, each with constant volume of 300 cm3 . The base of the
can is made from a material that costs 50 FRW per cm 2 , and the remaining
part is made of material that costs 20 FRW per cm 2 . Assume you are the
manager of the company, and you have to buy the material for constructing
the can. Which question do you ask yourself regarding the dimensions of
the can and the money to use for buying the material?

CONTENT SUMMARY
d  dy 
 >0
If function y = f ( x) i such that dy = 0 at x0 and dx  dx  at x0 , then
s dx
the function y = f ( x) has a local minimum at x0 ; the minimum value of the
function is f ( x0 ) .

dy
Function y = f ( x) is said to be increasing for the values of x such that > 0,
dx
dy
and decreasing for the values of x such that < 0,
dx
Example 3.2.1.

1 3 9 2
Given the total cost function P = Q − Q + 14Q + 22 , find the value of Q for
3 2
which the total cost is minimum, and find the minimum total cost

Solution:

dP
= Q 2 − 9Q + 14
dQ

dP
= 0 if and only if Q 2 − 9Q + 14 =
0 ; solving this quadratic equation, we get
dQ
Q1 = 2 and Q2 = 7 ;

Mathematics | Student Book | Senior Five | Experimental Version 65

 d
(Q 2 − 9Q + 14) = 2 Q− 9 ; its value at Q1 = 2 is 2(2) − 9 =−5 < 0 ;
dQ
At Q2 = 7 is 2(7) − 9 = 5 > 0 ;

Therefore, the minimum value of the total cost occurs for Q = 7 , and the

1 3 9 2
corresponding total cost is (7) − (7) + 14(7) + 22 =
13.83
3 2

Application activity 3.2.1

Find the value of Q for which the total cost is minimum, and find the
minimum total cost in each of the following cases:
a) P= ln(Q 2 − 8Q + 20)
1 3 17 2
b) P = Q − Q + 60Q + 27
3 2

3.2.2. Maximization of the total revenue function

Learning Activity 3.2.1

Consider the following problem: A company has to buy a plot for the building
of its factory. The plot must have a rectangular shape with a constant
perimeter of 400 meters, and the cost of the plot is constant. Assume you
are the manager of the company, and you have to choose the dimensions of
the rectangular plot located in a flat uniform area. Which question do you
ask yourself regarding the dimensions of the plot and the area of the plot?

CONTENT SUMMARY

If the total cost function is y = f ( x) , t en the total revenue function is R = xy .

h
dR
=0 d  dR 
Suppose dx at x0 and   < 0 at x0 , then the total revenue function
dx  dx 
R = xy has a local maximum at x0 ; the maximum value of the total revenue
function is R( x0 ) .

66 Mathematics | Student Book | Senior Five | Experimental Version

Example 3.2.2.

Given the total cost function, P = 5e −0.2Q , find the value of Q for which the
total revenue is Given the demand function maximum, and find the maximum
revenue.
Solution:

dR
The total revenue is R(Q) = 5Qe −0.2Q . Then, = (5 − Q)e −0.2Q ;
dQ

dR d
= 0 if and only if Q = 5 ; [(5 − Q)e −0.2Q ] = (−2 + 0.2Q)e −0.2Q ;
dQ dQ

1
its value at Q = 5 is −e −1 =− < 0 ;
e
Therefore, the maximum value of the total revenue occurs for Q = 5 and the

25
maximum value is R(5)
= = 9.1
e

Application activity 3.2.2

Given the demand function, P= 24 − 3Q , find the value of Q at which the

total revenue is maximum, and find the maximum revenue.

3.3. Price elasticity

3.3.1. Elasticity of demand

Learning Activity 3.3.1

The price P of a commodity and the quantity demanded Qd are related by

the equation. Qd = f ( P) . It is observed that P increases, for a particular
value of P . Determine whether Qd will decrease or increase, or neither.
a) Determine the percentage of decrease or increase

dQd P dQ P
b) Calculate . , for P = 10 . Predict a name for d .
dP Qd dP Qd

Mathematics | Student Book | Senior Five | Experimental Version 67

CONTENT SUMMARY

In Economics, price elasticity ε d measures the percentage change in quantity

associated with a percentage change in price. If the quantity Qd is related to price
dQ P
εd = d .
P by, Qd = f ( P) , then the elasticity of demand s defined by dP Qd .
i
Price elasticity of demand indicates how consumers respond to the change in
the amount proposed by the producers.

If ε d < 0 , then Qd and P are such that the increase in P implies the decrease in
Qd

Example 3.3.1.

The demand Qd is related to the price P by the function, Qd = 650 − 5 P − P 2 .

Find the price elasticity of the demand at P = 10

Solution:

dQd
=−5 − 2 P ; Elasticity of the demand:
dP

dQd P 10
εd = . =[−5 − 2(10)] =−0.5
dP Qd 650 − 5(10) − (10) 2

Application activity 3.3.1

Find the price elasticity of the demand if the quantity demanded Qd and
the price P are related by:

100
a) Qd ln=
=
P2
;P 4

20
b) Qd
= = ;P 3
P +1

68 Mathematics | Student Book | Senior Five | Experimental Version

 3.3.2. Elasticity of supply

Learning Activity 3.3.2

A beverage company estimates that the amount Qs of soft drink supplied per
month and the quantity C bought by customers are related by a function
Qs = f (C ) .It is observed that C increases, for a particular value of C .

Determine whether Qs will decrease or increase, or neither;

CONTENT SUMMARY

If the quantity Qs is related to price P by Qs = f ( P) , then the elasticity of

dQ P
εs = s .
deman is defined by dP Qs . In some cases, P is gi en in terms of Qs .
v
d
In this case, start by making Qs the subject of the formula.

dQs P
The price elasticity of supply is defined by, ε s = . ,
dP Qs
Where Qs : quantity supplied, and P : amount received from consumers.
Price elasticity of supply indicates how producers respond to the change in the
amount they receive from the consumers

Example 3.3.2.

The quantity supplied by the producers Qs is related to the amount of money P

received from consumers by the function P =−2 + 5Qs .Find the price elasticity
of supply for P = 3

Solution:

1 2 dQs 1 1 2
We have: Q
=s P+ ; = ; For P = 3 , Qs= (3) + = 1
5 5 dP 5 5 5

dQs P 1 3 3
εs
The price elasticity of supply is= .= =. ,
dP Qs 5 1 5

Mathematics | Student Book | Senior Five | Experimental Version 69

 Application activity 3.3.1

Find the price elasticity of the supply if the quantity supplied Qs and the
price P are related by:
a) P= 3 + 4Qs ; P = 11

Pe −0.2 P ; P 4
b) Qs 5=
=

End of unit assessment 3

1. Find the price and the quantity that will maximize the total revenue,
given the demand function: P = 12.5e −0.005Q

4x
2. Find, the coordinates of the minimum point of function, f ( x) =
, and confirm that it is a minimum. 3ln x

3. Given the total cost function, C =Q 3 − 3Q 2 + 15Q , find the marginal

cost.

4. = 30 − Q , find the marginal total revenue.

Given the cost function, C

5. Q 150 − 15 P at
Find the price elasticity of the demand function =
P = 4 , where P is the price.

70 Mathematics | Student Book | Senior Five | Experimental Version

 4
Univariate Statistics and
Unit Applications

Key Unit competence: Apply univariate statistical concepts to collect, organise,

analyse, interpret data, and draw appropriate decisions

Introductory activity

1. (a) How do you think collecting and keeping data is important

daily?
b) In your field of study, which kind of data can a person collect?
And give an example of each kind of data.
c) Is collecting, organizing, and interpreting data helpful in making
a family budget? What do you think about a national budget?
2. Suppose you have a shop selling food, and you want to know the
type of food most people prefer to buy:
i) Which statistical information will you need to collect?
ii) How will you collect such information?
iii) Which statistical measure will help you know the most
preferred food?
3. During an accounting exam, out of 10, ten students scored the
following marks: 3, 5, 6, 3, 8, 7, 8, 4, 8, 6.
a) Determine the mean mark of the class.
b) What is the mark that many students obtained?
c) Compare and discuss the mean mark of the class and the mark
for every student. What advice could you give to an accounting
teacher?

Mathematics | Student Book | Senior Five | Experimental Version 71

4.1 Basic concepts in univariate statistics
4.1.1. Statistical concepts

Learning Activity 4.1.1

1. Using the internet or any other resources, do research.
i) What do you understand by the term statistics?
ii) What are the different branches of statistics?
iii) What are the key terms used in statistics?
2. Suppose your company is given a market for supplying milk and
fruits to all primary school students in Rwanda. If the student must
choose between milk and fruits, what should a company do to ensure
that it will supply what students want?

CONTENT SUMMARY
Statistics is the branch of mathematics that deals with data collection,
data organization, summarization, analysis, interpretation, and drawing of
conclusions from numerical facts or data.
Statistics plays a vital role in nearly all businesses and forms the backbone for
all future development strategies. Every business plan starts with extensive
research, which is all compiled into statistics that can influence a final decision.
Statistics helps the businessman to plan production according to the taste of
customers.

Branches of statistics
There are two branches of statistics, namely descriptive, and inferential
statistics.
a. Descriptive statistics
Descriptive statistics deals with describing the population under study. It
consists of the collection, organization, summarization, and presentation of
data in a convenient and usable form.

Examples of descriptive statistics

• The average score of accounting students on the mathematics test.
• The average monthly salary of the employees in a company.
• The average age of the people who voted for the winning candidate in the
last election.

72 Mathematics | Student Book | Senior Five | Experimental Version

b. Inferential statistics
Inferential statistics consists of generalizing from samples to populations,
performing estimations and hypothesis tests, determining relationships among
variables, and making predictions.
The results of the analysis of the sample can be deduced to the larger population
from which the sample is taken. It consists of a body of methods for drawing
conclusions or inferences about characteristics of a population based on
information contained in a sample taken from the population. This is because
populations are almost very large; investigating each member of the population
would be impractical and expensive.

Examples of inferential statistics

• Collecting the monthly savings data of every family that constitutes your
population may be challenging if you are interested in the savings pattern
of an entire country. In this case, you will take a small sample of families
from across the country to represent the larger population of Rwanda. You
will use this sample data to calculate its mean and standard deviation.
• Suppose you want to know the percentage of people who love shopping
at SIMBA supermarket. We take the sample of the population and find
the proportion of individuals who love the SIMBA supermarket. With the
assistance of probability, this sample proportion allows us to make a few
assumptions about the population proportion.
In statistics, we generally want to study a population. Because it takes a lot
of time and money to examine an entire population, we select a sample to
represent the whole population.
The population is a collection of persons, things, or objects under the study.
The population is also defined as the universe, or the entire category under
consideration.
A sample is the portion of the population that is available, or to be made
available, for analysis. A sample is also defined as a subset of the population
studied. From the sample data, we can calculate a statistic.
A statistic is a number that represents a property of the sample. For example, if
we consider one district in Rwanda to be a sample of the population of districts,
then the average (mean) income generated by that one district at the end of
the financial year is an example of a statistic. The statistic is an estimate of a
population parameter, in this case the mean.
A parameter is a numerical characteristic of the whole population that can be
estimated by a statistic. Since we considered all districts to be the population,

Mathematics | Student Book | Senior Five | Experimental Version 73

then, the average (mean) income generated by district over the entire district
is an example of a parameter.

Application activity 4.1.1

1. Using an example, differentiate descriptive statistics from inferential
statistics.
2. We want to know the average (mean) amount of money senior five
students spend at Kiziguro secondary school on school supplies
that do not include books. We randomly surveyed 100 first-year
students at the school. Three of those students spent 1500Frw,
2000Frw, and 2500Frw, respectively. In this example, what could
be the population, sample, statistic and parameter?

4.1.2 Variables and types of variables

Learning Activity 4.1.1

Gisubizo conducted research on clients’ satisfaction with bank services.
She wanted to understand the relationship between clients’ satisfaction
and the amount of money they saved in that bank.
a) What could be the variables to consider in her research?
b) Will those variables give qualitative or quantitative information?

CONTENT SUMMARY
A variable is a characteristic of interest for each person or object in a population.
A variable is a characteristic under study that takes different values for different
elements. A variable, or random variable, is a characteristic or measurement
that can be determined for each member of a population. For example, if we
want to know the average (mean) amount of money senior five students spend
at Kiziguro secondary school on school supplies that do not include books.
We randomly surveyed 100 first year students at the school. Three of those
students spent 1500Frw, 2000Frw, and 2500Frw, respectively. In this example,
the variable could be the amount of money spent (excluding books) by one
senior five student. Let X = the amount of money spent (excluding books) by
one senior five student attending Kiziguro secondary school. Another example,
if we collect information about income of households, then income is a variable.
These households are expected to have different incomes; also, some of them
may have the same income.

74 Mathematics | Student Book | Senior Five | Experimental Version

Note that a variable is often denoted by a capital letter like X , Y, Z ,.... and
their values denoted by small letters for example x, y, z ,.... The value of a
variable for an element is called an observation or measurement.
In statistics, we can collect data on a single variable or many variables. For
example, if we are interested in knowing how well the company is paying its
employees, we shall only collect data on the salaries of the workers in the
company. In this case, we will categorize these statistics as univariate statistics.
This unit only discusses univariate statistics and its application. When one
variable causes change in another, we call the first variable the independent
variable or explanatory variable. The affected variable is called the dependent
variable or response variable. There are mainly two types of variables:
qualitative variables and quantitative variables.
• Qualitative variables
Qualitative variables are variables that cannot be expressed using a number.
They express a qualitative attribute, such as hair color, religion, race, gender,
social status, method of payment, and so on. The values of a qualitative variable
do not imply a meaningful numerical ordering.
Qualitative variables are sometimes referred to as categorical variables.
For example, the variable sex has two distinct categories: ‘male’ and ‘female.’
Since the values of this variable are expressed in categories, we refer to this as
a categorical variable. Similarly, the place of residence may be categorized as
urban and rural and thus is a categorical variable. Categorical variables may
again be described as nominal and ordinal. Ordinal variables can be logically
ordered or ranked higher or lower than another but do not necessarily establish
a numeric difference between each category, such as examination grades (A+, A,
B+, etc., and clothing size (Extra large, large, medium, small). Nominal variables
are those that can neither be ranked nor logically ordered, such as religion, sex,
etc.
• Quantitative variables
Quantitative variables also called numeric variables, are those variables that
are expressed in numerical terms, counted or compared on a scale. A simple
example of a quantitative variable is a person’s age. Age can take on different
values because a person can be 20 years old, 35 years old, and so on. Likewise,
family size is a quantitative variable because a family might be comprised of one,
two, or three members, and so on. Each of these properties or characteristics
referred to above varies or differs from one individual to another. Note that
these variables are expressed in numbers, for which we call quantitative or
sometimes numeric variables. A quantitative variable is one for which the
resulting observations are numeric and thus possess a natural ordering or
ranking.

Mathematics | Student Book | Senior Five | Experimental Version 75

Quantitative variables are again of two types: discrete, and continuous. Variables
such as some children in a household or the number of defective items in a box
are discrete variables since the possible scores are discrete on the scale. For
example, a household could have three or five children, but not 4.52 children.
Other variables, such as ‘time required to complete a test’ and ‘waiting time in a
queue in front of a bank counter,’ are continuous variables. The time required
in the above examples is a continuous variable, which could be, for example,
1.65 minutes or 1.6584795214 minutes.

Application activity 4.1.2

Suppose you have a company that sells electronic devices.
i) If you are interested in understanding how your clients are satisfied
with your products. Which variable (s) will you consider in collecting
the data? Is this a univariate statistics? Why?
ii) If you are interested in understanding the relationship between
clients’ satisfaction and educational levels. Which variable (s) will
you consider in collecting the data? Is this a univariate statistics?
Why?

4.1.3 Data and types of data

Learning Activity 4.1.3

Using the internet or any other resources, do research. What do you
understand by the term data? Give an example.

CONTENT SUMMARY
Data are individual items of information that come from a population or sample.
Data is also defined as a set of observations. Data are the values (measurements
or observations) that the variables can assume. They may be numbers, or they
may be words. Datum is a single value. Data may come from a population or
from a sample. Lower case letters like x or y generally are used to represent
data values.
The observations or values that differ significantly from others are called
outliers. Outliers are at the extreme ends of a dataset. Dataset is a collection
of data of any particular study without any manipulation. Information are
facts about something or someone. Most data can be put into the following
categories: qualitative, and quantitative.

76 Mathematics | Student Book | Senior Five | Experimental Version

Qualitative data are the result of categorizing or describing attributes of a
population. Qualitative data are also often called categorical data. Clients’
satisfaction, quality of goods, color of the car a person bought are some
examples of qualitative (categorical) data. Qualitative (categorical) data are
generally described by words or letters.
Quantitative data are the result of counting or measuring attributes of a
population. Quantitative data are always numbers. Amount of money, number
of items bought in a supermarket, and numbers of employees of the company
are some examples of quantitative data. Quantitative data may be either
discrete or continuous. Data is discrete if it is the result of counting (such
as the number of students of a given gender in a class or the number of books
on a shelf). Data is continuous if it is the result of measuring (such as distance
traveled or weight of luggage). All data that are the result of counting are called
quantitative discrete data.
These data take on only certain numerical values. If you count the number of
phone calls you receive for each day of the week, you might get values such as
zero, one, two, or three. Data that are not only made up of counting numbers,
but that may include fractions, decimals, or irrational numbers, are called
quantitative continuous data.

Example
You go to the supermarket and purchase three soft drinks (500ml soda, 1ml
milk and 300ml juice) at 5000frw, four different kinds of fruits (apple, mango,
banana and avocado) at 800frw, two different kinds of vegetables (broccoli and
carrots) at 500frw, and two desserts (ice cream and biscuits) at 1000frw.
In this example,
• Number of soft drinks, different kinds of fruits, different kinds of vegetables,
and desserts purchased are quantitative discrete data because you count
them.
• The prices (5000frw, 800frw, 500frw, and 1000frw) are quantitative
continuous data.
• Types of soft drinks, vegetable, fruits, and desserts are qualitative or
categorical data.
A collection of information which is managed such that it can be updated
and easily accessed is called a database. A software package which can be
used to manipulate, validate and retrieve this database is called a Database
Management System. For example, Airlines use this software package to book
tickets and confirm reservations which are then managed to keep a track of the
schedule.

Mathematics | Student Book | Senior Five | Experimental Version 77

 Application activity 4.2.1
Describe the data and types of data used in the following study. We want
to know the average amount of money spent on school uniforms annually
by families with children at G.S Kayonza. We randomly survey ten families
with children in the school. Ten families spent 65000Frw, 45000Frw,
65000Frw, 15000Frw, 55000Frw, 35000Frw, 25000Frw, 45000Frw,
85000Frw and 95000Frw, respectively.

4.1.4. Levels of measurement scale

Learning Activity 4.1.4

A researcher surveyed 100 people and asked them what type of place
they visited (rural or urban) and how satisfied (very satisfied, satisfied,
somehow satisfied, not satisfied) they were with their most recent visit to
that place. Those people were also asked to provide their ages. What are
the variables involved in this research?
Classify those variables according to how their data values could be
categorized or measured. Is it possible to rank data values obtained from
those variables?
If yes, rank them. Is it possible to find the difference between the data
values of each variable?

CONTENT SUMMARY
Variables classified according to how they are categorized or measured. For
example, the data could be organized into specific categories, such as major
field (accounting, finance, etc.), nationality or gender. On the other hand, can the
data values could be ranked, such as grade (A, B, C, D, F) or rating scale (poor,
good, excellent), or they can be classified according to the values obtained from
measurement, such as temperature, heights, or weights.
Therefore, we need to distinguish between them through the measurement
scale used. A scale is a device or an object used to measure or quantifies any
event or another object. In statistics, the variables are defined and categorized
using different levels of measurements. Level of measurement or scale of
measure is a classification that describes the nature of data within the values
assigned to variables (Kirch, 2008).

78 Mathematics | Student Book | Senior Five | Experimental Version

There are four levels or scales of the measurement: Nominal, Ordinal, Interval,
and Ratio.

Nominal scale
A nominal scale is used to name the categories within the variables by providing
no ranking or ordering of values; it simply provides a name for each category
within a variable so that you can track them among your data (Crossman, 2020).
The nominal level of measurement is also known as a categorical measure and
is considered qualitative in nature.

Examples
• Nominal tracking of gender (male or female)
• Nominal tracking of travel class (first class, business class and economy
class).
When the classification takes ranks into consideration, the ordinal level of
measurement is preferred to be used.

Ordinal scale
The ordinal level of measurement classifies data into categories that can be
ordered, however precise differences between the ranks do not exist. Ordinal
scales are used when people want to measure something that is not easily
quantified, like feelings or opinions. Within such a scale the different values
for a variable are progressively ordered, which is what makes the scale useful
and informative. However, it is important to note that the precise differences
between the variable categories are unknowable. Ordinal scales are commonly
used to measure people’s views and opinions on social issues, like quality of the
products, services, or how people are satisfied with something.

Examples
• if you have a business and you wish to know how people are happy with
your products or services, you could ask them a question like “How happy
are you with our products or services?” and provide the following response
options: “Very happy,” “Somehow happy,” and “Not happy.”
• To test the quality of the canned product, people can use the rating scale
either excellent or good or bad.

Interval scale
Unlike nominal and ordinal scales, an interval scale is a numeric one that allows
for ordering of variables and provides a precise, quantifiable understanding of

Mathematics | Student Book | Senior Five | Experimental Version 79

the differences between them.

Example
It is common to measure people’s income as a range like 0Frw-100,000Frw;
100,001Frw-200,000Frw; 200,001Frw-300,000Frw, and so on. These ranges
can be turned into intervals that reflect the increasing level of income, by using
1 to signal the lowest category, 2 the next, then 3, etc.

Ratio scale
The ratio scale is the interval level with additional property that there is also
a natural zero starting point. In this type of scale zero means nothingness.
Another difference lies in that we can attribute some of the quantities to others.

Example
The value of salary for someone is a measurement of type ratio level, where
we can attribute values of wages to each other, as if to say that the person X
receives a salary twice the salary of the person Y. And zero here means that the
person did not receive a salary.

Application activity 4.2.1

Classify each according to the level of measurement with the interpretation
of the meaning of zero if it exists.
i) Ages of the company workers (in years).
ii) Color of clothes in a shop.
iii) Temperatures inside the room (in Celsius).
iv) Nationalities of the company workers.
v) Salaries of the company employees.
vi)Weights of boxes of fruits

4.1.5. Sampling and Sampling methods

80 Mathematics | Student Book | Senior Five | Experimental Version

 Learning Activity 4.1.5
Suppose that a certain Secondary School has 10,000 boarding students
(the population).
We are interested in the average amount of money a boarding student
spends on meals and accommodation in the year. Asking all 10,000 students
is almost an impossible task. What would you advise that school to do so
that it gets the needed information to know the average amount of money
students are spending?
How will it be done so that the information the school gets represents the
population?

CONTENT SUMMARY
Collecting data on entire population is costly or sometimes impossible.
Therefore, a subset or subgroup of the population can be selected to represent
the entire population. The process of selecting a sample from an entire
population is called sampling. Since the sample selected is representing the
wholepopulationunderstudy,thesamplemusthavethesamecharacteristicsas
the population. There are several ways of selecting sample from the population.
Some of the methods used in selecting samples are simple random sampling,
stratified sampling, cluster sampling, and systematic sampling.
In stratified sampling, the population is divided into groups called strata
and then takes a proportionate number from each stratum. For example, you
can stratify (group) taxpayers by their Ubudehe categories then choose a
proportionate simple random sample from each stratum (Ubudehe category)
to get a stratified random sample. To choose a simple random sample from each
category, number each member of the first category, number each member
of the second category, and do the same for the remaining categories. Then
use simple random sampling to choose proportionate numbers from the first
category and do the same for each of the remaining categories. Those numbers
picked from the first category, picked from the second category, and so on
represent the members who make up the stratified sample.
In cluster sampling, the population is divided the population into clusters
(groups) and then randomly select some of the clusters. All the members from
these clusters are in the cluster sample.
For example, if you randomly sample your costumers by gender (males,

Mathematics | Student Book | Senior Five | Experimental Version 81

females, those who prefer not to say), the three groups make up the cluster
sample. Number each group, and then choose four different numbers using
simplerandomsampling.Allmembersofthethreegroupswiththosenumbers
are the cluster sample.
In systematic sampling, we randomly select a starting point and take
every nth piece of data from a listing of the population.
For example, suppose you have to do a phone survey. Your phone book contains
20,000 customers listings. You must choose 400 names for the sample. Number
the population 1–20,000 and then use a simple random sample to pick a
number that represents the first name in the sample. Then choose every fiftieth
name thereafter until you have a total of 400 names (you might have to go back
to the beginning of your phone list). Systematic sampling is frequently chosen
because it is a simple method. All the above-mentioned sampling methods are
random.
A type of sampling that is non-random is convenience sampling. Convenience
sampling involves using results that are readily available.
For example, a computer software store conducts a marketing study by
interviewing potential customers who happen to be in the store browsing
through the available software. The results of convenience sampling may be
very good in some cases and highly biased (favor certain outcomes) in others.

Application activity 4.1.5

A school account conducted a study to determine the average school fees
parents pay yearly. Each parent in the following samples is asked how
much fee he or she paid for each term. What is the type of sampling in each
case?
a) A random number generator is used to select a parent from the
alphabetical listing of all parents. Starting with that student,
every 50th parent is chosen until 75 parents are included in the
sample.
b) A completely random method is used to select 75 parents. Each
parent has the same probability of being chosen at any stage of
the sampling process.
c) The parents who have students in nursery, primary, and
secondary are numbered one, two, and three, respectively. A
random number generator is used to pick two of those years. All
students in those two years are in the sample.

82 Mathematics | Student Book | Senior Five | Experimental Version

 d) A sample of 100 parents having students at a school is taken
by organizing the parents’ names by classification as a nursery
(parents whose kids are in the nursery), junior (parents whose
kids are in primary), or senior (parents whose kids are in
secondary), and then selecting 25 parents from each.
e) An accountant is requested to ask the first ten parents he
encounters outside the school what they paid for tuition fees.
Those ten parents are the sample.

4.2 Organizing and graphing data

4.2.1. Frequency table

Learning Activity 4.2.1

The weekly revenues paid (in Frw) by 20 businesspeople are below. 27000,
31000, 24000, 31000, 26000, 36000, 21000, 22000, 34000, 29000, 25000,
29000, 27000, 39000, 27000, 23000, 28000, 29000, 24000, 27000. Which
revenue has been paid by many people? Represent this data in a tabular
form (revenue and the number of people who paid each revenue).

CONTENT SUMMARY
Frequency tables are a great starting place for summarizing and organizing
your data. Once you have a set of data, you may first want to organize it to see
the frequency, or how often each value occurs in the set. Frequency tables can
be used to show either quantitative or categorical data.
Example
Assume that a sample of 50 taxpayers in a district was selected to understand
how taxpayers are satisfied with the taxes they are paying. The responses of
those taxpayers are recorded below where (v) means very high satisfied, (s)
means somewhat satisfied and (n) means not satisfied. v, n, v, n, v, s, n, n, n, n, s,
s, v, n, n, n, s, n, n, s, n, n, n, s, s, s, v, v, s, v, s, v, n, n, n, n, s, v, v, v, v, v, v, v, s, v, v, v, v, v
From the recorded data above, we note that:
• Eleven of them were not satisfied with the taxes they were paying.
• Five of them were somewhat satisfied with the taxes were paying.
• Four of them were very high satisfied with the taxes were paying.

Mathematics | Student Book | Senior Five | Experimental Version 83

This information can be presented in a tabular form which lists the type of
satisfaction (very high satisfied, somewhat satisfied, and not satisfied) and the
number of students corresponding to each category. Clearly the variable is the
type of satisfaction, which is qualitative variable.
Note that, each of the students belongs to one and only one of the categories.
The number of students who belong to a certain category is called the frequency
of that category. A frequency table shows how the frequencies are distributed
over various categories.
Table 4.1: Frequency table

Type of satisfaction (variable) Number of student (Frequency)

Very high satisfied (v) 20

Somewhat satisfied (s) 12

Not satisfied (n) 18

Sum=50

Application activity 42.1

Consider the data on the marital status of 50 people who were interviewed.

Married Married Married Married

Married Divorced
Single
Separated Separated Separated Separated Separated Divorced
Single
Single Single Single Single Single Divorced

Single Single Married Single Separated Divorced

Single Separated Married Married Divorced Divorced

Divorced Single Single Married Divorced Single

Separated Single Single Single Married Married

Separated Single Single Single Single Separated

Represent the above data in a frequency table.

84 Mathematics | Student Book | Senior Five | Experimental Version

 4.2.2. Bar graph

Learning Activity 4.2.1

August 27, 1991, Wall Street Journal (WSJ) article reported that the
industry’s biggest companies are absorbing increasing numbers of small
software firms. According to WSJ, the result of this dominance by a few
giants is that the industry has become tougher for software entrepreneurs to
break into. The newspaper printed the chart in the accompanying figure to
depict software companies’ market share breakdown. From entrepreneurs
to corporate giants: market share among the top 100 software companies,
based on total 1990 revenue of $5.7 billion. Refer to this chart to answer
the following questions:
a) List the companies in descending order of market share.
b) What is the combined market share for Lotus Development and
WordPerfect?
c) What is the combined market share for Micro soft, Lotus
Development, and Novell?

CONTENT SUMMARY
A bar chart or bar graph is a chart or graph that presents numerical data with
rectangular bars with heights or lengths proportional to the values that they
represent. The bars can be plotted vertically or horizontally. A vertical bar chart
is sometimes called a line graph.
To construct a bar graph, we use the following steps:
• Represent the categories on the horizontal axis (remember to represent all
categories with equal intervals).
• Mark the frequencies (or percentages) on the vertical axis.
• Draw one bar for each category that corresponds to its frequency (or
percentage) on the vertical axis.

Mathematics | Student Book | Senior Five | Experimental Version 85

Example 1
The table below shows number of learners per class in a certain school in
Rwanda.

Class S1 S2 S3 S4 S5 S6

Number of learners 45 40 50 60 55 45

a) Represent the data in a bar chart

b) How many learners are in the whole school?

Solution
a) A bar graph showing number of learners per class in a school

b) The number of learners that are in the whole school

= 45 + 40 + 50 + 60 + 55 + 45 = 295

The school has 295 learners.

Exampe2
An insurance company determines vehicle insurance premiums based on
known risk factors. If a person is considered a higher risk, their premiums will
be higher. One potential factor is the color of your car. The insurance company
believes that people with some color cars are more likely to get in accidents. To
research this, they examine police reports for recent total loss collisions. The
data is summarized in the frequency table below:

86 Mathematics | Student Book | Senior Five | Experimental Version

Color Frequency
Blue 25
Green 52
Red 41
White 36
Black 39
Grey 23
a) From the frequency table above, identify the highest frequency and
the lowest frequency.
b) Present the car data on bar chart indicating frequency against vehicle
color involved in total loss collision.

Solution
a) From the bar chart, the highest frequency is 52 and the lowest
frequency is 23
b) The bar chart indicating frequency against vehicle color involved in
total loss collision

Mathematics | Student Book | Senior Five | Experimental Version 87

 Application activity 42.2
1. Iyamuremye is approaching retirement with a portfolio consisting
of cash and money market fund investments worth 1,350,000,
bonds worth 1,650,000, stocks worth 1,850,000, and real estate
worth 12,000,000. Present these data in a bar chart.
2. By the end of 2022, MTN Rwanda had over 5 million users. The table
below shows three age groups, the number of users in each age
group, and the proportion (%) of users in each age group. Construct
a bar graph of this data.

Age groups Number of MTN users Proportion (%) of MTN

users
13–25 2,250,000 45%

26–44 1,800,000 36%

45–64 950,000 19%

4.2.3 Histogram and polygon

Learning Activity 4.2.3

The graph below indicates the number of hours people work during a week.
The vertical axis represents the number of people, while the horizontal axis
represents the number of hours people spend at work.
a) How many people spend more hours at work? How many hours
do those people spend?
b) In total, how many people participated in this study?
c) What is the name of this graph?

88 Mathematics | Student Book | Senior Five | Experimental Version

CONTENT SUMMARY
After you have organized the data into a frequency distribution, you can
present them in graphical form. The purpose of graphs in statistics is to
convey the data to the viewers in pictorial form. It is easier for most people to
comprehend the meaning of data presented graphically than data presented
numerically in tables or frequency distributions.
The three most commonly used graphs in research are
a) The histogram.
b) The frequency polygon.
c) The cumulative frequency graph or ogive (pronounced o-jive).
a. The Histogram
The histogram is a graph that displays the data by using contiguous vertical
bars (unless the frequency of a class is 0) of various heights to represent the
frequencies of the classes.
Example1: suppose the age distribution of personnel at a small business is:
25, 24, 29, 20, 32, 39, 36, 30, 30, 39, 40, 42, 45, 47, 48, 43, 49, 50, 54, 58, 50,
65, 79.
Form classes by grouping ages of these personnel in categories as follows: 20-
29, 30-35, 39, 40-49, 50-59, 60-69, 70-79.
For each group, write the number of times numbers in that group are
occurring. To construct a histogram, we need to enter a scale on the horizontal
axis. Because the data are discrete, there is a gap between the class intervals,
say between 20 and 29 and 30–39.
In such a case, we will use the midpoint between the end of one class and the
beginning of the next as our dividing point. Between the 20–29 interval and

30–39 interval, the dividing point will be

( 20 + 29 ) = 19.5 , ( 29 + 30 ) = 29.5
2 2
respectively. We find the dividing point between the remaining classes
similarly.

Mathematics | Student Book | Senior Five | Experimental Version 89

Example2: Construct a histogram to represent the data shown for the record
high temperatures for each of the 50 states.

Class boundaries Frequency

99.5-104.5 2
104.5-109.5 8
109.5-114.5 18
114.5-119.5 13
119.5-124.5 7
124.5-129.5 1
129.5-134.5 1

Step 1: Draw and label the x and y axes. The x axis is always the horizontal axis,
and the y axis is always the vertical axis.
Step 2: Represent the frequency on the y axis and the class boundaries on the
x axis.
Step 3: Using the frequencies as the heights, draw vertical bars for each class.
See Figure below

90 Mathematics | Student Book | Senior Five | Experimental Version

b. The Frequency Polygon
The frequency polygon is a graph that displays the data by using lines that
connect points plotted for the frequencies at the midpoints of the classes. The
frequencies are represented by the heights of the points.

Example:
Using the frequency distribution given in Example 2, construct a frequency
polygon
Step 1: Find the midpoints of each class. Recall that midpoints are found by
adding the upper and lower boundaries and dividing by 2:

99.5 + 104.5 104.5 + 109.5

= 102 , = 107 and so on.
2 2
The midpoints are:

Step 2: Draw the x and y axes. Label the x axis with the midpoint of each class,
and then use a suitable scale on the y axis for the frequencies.

Mathematics | Student Book | Senior Five | Experimental Version 91

Step 3: Using the midpoints for the x values and the frequencies as the y values,
plot the points.
Step 4: Connect adjacent points with line segments. Draw a line back to the
x axis at the beginning and end of the graph, at the same distance that the
previous and next midpoints would be located, as shown in figure.

The frequency polygon and the histogram are two different ways to represent
the same data set. The choice of which one to use is left to the discretion of the
researcher.
c. The cumulative frequency graph or Ogive.
The ogive is a graph that represents the cumulative frequencies for the classes
in a frequency distribution.
Step 1: Find the cumulative frequency for each class.

Step 2: Draw the x and y axes. Label the x axis with the class boundaries. Use
an appropriate scale for the y axis to represent the cumulative frequencies.

92 Mathematics | Student Book | Senior Five | Experimental Version

(Depending on the numbers in the cumulative frequency columns, scales such
as 0, 1, 2, 3, . . . , or 5, 10, 15, 20, . . . , or 1000, 2000, 3000, . . . can be used.
Do not label the y axis with the numbers in the cumulative frequency column.)
In this example, a scale of 0, 5, 10, 15, . . . will be used.
Step 3: Plot the cumulative frequency at each upper class boundary, as shown
in Figure below. Upper boundaries are used since the cumulative frequencies
represent the number of data values accumulated up to the upper boundary of
each class.

Step 4: Starting with the first upper class boundary, 104.5, connect adjacent
points with line segments, as shown in the figure. Then extend the graph to the
first lower class boundary, 99.5, on the x axis.

Mathematics | Student Book | Senior Five | Experimental Version 93

Cumulative frequency graphs are used to visually represent how many values
are below a certain upper class boundary. For example, to find out how many
record high temperatures are less than 114.5 0F, locate 114.5 0F on the x axis,
draw a vertical line up until it intersects the graph, and then draw a horizontal
line at that point to the y axis. The y axis value is 28, as shown in the figure.

Application activity 42.3

Consider the following data:
45,50,55,60,65,70,75,47,51,56,61,66,71,76,48,52,57,62,67,72,77,49,53,5
8,63,68,73,78,49,54,59,64,68,74,49,51,55,61,68,71,51,56,61,69,71,52,56,
62,66,72,53,57,62,67,72,54,58, 63,67,74,58,63,68,58,64,68,59,64,69,55,6
4,69,56,64,68,61, 61,62,62,63.
Then:
a) Make this data in a frequency distribution table with Class
boundaries width equal to 5 and containing Class boundaries,
Midpoints, Frequencies, Relative frequencies, Percentages, and
Cumulative frequencies.
b) Draw the histogram for the frequencies, relative frequencies,
percentages, and percentage frequencies in the distribution
table.

94 Mathematics | Student Book | Senior Five | Experimental Version

 4.2.4 Time series graph

Learning Activity 4.2.4

The graph below shows the number of buyers per quarter (per three
months) who have visited a supermarket.

From the graph,

a) In which quarter did few people visit the supermarket?
b) How many people did buy at the supermarket in the second
quarter of 2005?
c) In total, how many buyers did visit the supermarket from 2005
to 2007?

CONTENT SUMMARY
In most graphs and charts, the independent variable is plotted on the horizontal
axis (the X − axis ) and the dependent variable on the vertical axis (the Y − axis ).
A time series is defined as having the independent variable of time and the
dependent variable as the value of the variable being studied.
A time series graph is a line graph that shows data such as measurements,
sales or frequencies over a given time.
Frequently, “time” is plotted along the x-axis. Such a graph is known as a time-
series graph because on it, changes in a dependent variable (such as GDP: Gross
Domestic Production, inflation rate, or stock prices) can be traced over time.
They can be used to show a pattern or trend in the data and are useful for
making predictions about the future such as weather forecasting or financial
growth.

Mathematics | Student Book | Senior Five | Experimental Version 95

To create the time series graph,
• Start off by labeling the time-axis in chronological order.
• Label the vertical axis and horizontal axis. The horizontal axis always
shows the time, and the vertical axis represents the variable being
recorded against time.
• After labelling, plot the points given in the data set.
• Finish the graph by connecting the dots with straight lines.

Example
In a week, a certain company is making a profit of 10000 FRW on the first
day, 15000FRW on the second day, 12000FRW on the third day, 13000FRW
on the fourth day, 9000FRW on the fifth day, 10000FRW on the sixth day, and
9000FRW on the seventh day. In a tabular form, this can be presented as

Day 1 2 3 4 5 6 7

P r o f i t 10000 15000 12000 13000 9000 10000 9000

(FRW)

The time series graph is

96 Mathematics | Student Book | Senior Five | Experimental Version

 Application activity 4.2.4
The following data shows the Annual Consumer Price Index each month
for ten years. Construct a time series graph for the Annual Consumer Price
Index data only.

Year Aug Sep Oct Nov Dec Annual

2003 184.6 185.2 185.0 184.5 184.3 184.0

2004 189.5 189.9 190.9 191.0 190.3 188.9

2005 196.4 198.8 199.2 197.6 196.8 195.3

2006 203.9 202.9 201.8 201.5 201.8 201.6

2007 207.917 208.490 208.936 210.177 210.036 207.342

2008 219.086 218.783 216.573 212.425 210.228 215.303

2009 215.834 215.969 216.177 216.330 215.949 214.537

2010 218.312 218.439 218.711 218.803 219.179 218.056

2011 226.545 226.889 226.421 226.230 225.672 224.939

2012 230.379 231.407 231.317 230.221 229.601 229.594

4.2.5 Pie chart

Learning Activity 4.2.5

The chart below presents data on why teenagers drink. Use the information
shown in the chart to answer the following questions:
a) For what reason do the highest numbers of teenagers drink?
b) What percentage of teenagers drink because they are bored or
upset?

Mathematics | Student Book | Senior Five | Experimental Version 97

CONTENT SUMMARY
A pie chart, sometimes called a circle chart, is a way of summarizing a set of
data in circular graph. This type of chart is a circle that is divided into sections
or wedges according to the percentage of frequencies in each category of the
distribution. Each part is represented in degrees.
To present data using pie chart, the following steps are respected:
Step 1: Write all the data into a table and add up all the values to get a total.
Step 2: To find the values in the form of a percentage divide each value by
the total and multiply by 100. That means that each frequency must also be
converted to a percentage by using the formula
f
%= .100%
n

Step 3: To find how many degrees for each pie sector we need, we take a full
circle of 360° and use the formula:

Frequency of S × 3600
Angle for sec tor S =
Total frequency

Since there are 3600 in a circle, the frequency for each class must be converted
into a proportional part of the circle. This conversion is done by using the
formula:
f
Degrees = .3600
n
where f is frequency for each class and n is the sum of the frequencies. Hence,
the following conversions are obtained. The degrees should sum to 360.
Step 4: Once all the degrees for creating a pie chart are calculated, draw a circle
(pie chart) using the calculated measurements with the help of a protractor,
and label each section with the name and percentages or degrees.

Example.
1. In the summer, a survey was conducted among 400 people about their
favourite beverages: 2% like cold-drinks, 6% like Iced-tea, 12% like
Cold-coffee, 24% like Coffee and 56% like Tea.
a) How many people like tea?
b) How many more people like coffee than cold coffee?

98 Mathematics | Student Book | Senior Five | Experimental Version

 c) What is the total central angle for iced tea and cold-drinks?
d) Draw a pie chart to represent the provided information.

Solution:
a) Total number of people = 400

56
Number of people like tea = 400 × = 224
100
24
a) Number of people like coffee = 400 × = 96
100
Number of people like cold-coffee = 400 ×12 = 48

Number of people like coffee more than cold-coffee = 96 − 48 = 48

6
a) Number of people like iced-tea = 400 × = 24
100

2
Number of people like cold-drinks = 400 × =8
100

24
Central angle for iced-tea = × 3600 =21.60
400

8
Central angle for cold-drinks = × 3600 =7.20 = 8/400 × 360o = 7.2o
400
Total central angle = 21.6 + 7.20 = 28.80 = 21.6o + 7.2o = 28.8o.
0

Mathematics | Student Book | Senior Five | Experimental Version 99

Example 2:
A person spends his time on different activities daily (in hours):

Activity Office exercise Travelling Watching Sleeping Miscellaneous

work shows
Number 9 1 2 3 7 2
of hours

a) Find the central angle and percentage for each activity.

b) Draw a pie chart for this information
c) Use the pie chart to comment on these findings.

Solution
a) Central angles are calculated by using the formula:

f
Degrees= × 3600 and percentages calculated by using the
n
f
formula: Percentage= ×100%
n

Number
Activity Central angle Percentage
of hours
9 9
Office work 9 × 3600 =
1350 ×100% =
37.5% ; 38%
24 24
1 1
Exercise 1 × 3600 =
150 ×100% =
4.16% ; 4%
24 24
2 2
Travelling 2 × 3600 =
300 ×100% =
8.33% ; 8%
24 24
Watching 3 3
3 × 3600 =
450 ×100% =
12.5% ; 13%
shows 24 24
7 7
Sleeping 7 × 3600 =
1050 ×100% =
29.16% ; 29%
24 24
2 2
Miscellaneous 2 × 3600 =
300 ×100% =
8.33% ; 8%
24 24
Total 24 3600 100

100 Mathematics | Student Book | Senior Five | Experimental Version

 b) Using a protractor, graph each section and write its name and
corresponding percentage, as shown in the Figure below

Application activity 4.2.5

After selling fruits in a market, Aisha had a total of 144 fruits remaining.
The pie chart below shows each type of fruit that remained.

a) Find the total cost of mangoes and pawpaws if a mango sells at

30 FRW and pawpaw at 160 FRW each.
b) Which types of fruit remained the most?
c) Draw a frequency table to display the information on the pie
chart.

Mathematics | Student Book | Senior Five | Experimental Version 101

 4.2.6 Graph interpretation

Learning Activity 4.2.4

The graph below shows the sizes of sweaters worn by 30 year 1 students
in a certain school. Observe it and interpret it by answering the questions
below:

a) How many students are with small size?

b) How many students with medium size, large size and extra large
size are there?

CONTENT SUMMARY
Once data has been collected, they may be presented or displayed in various
ways including graphs. Such displays make it easier to interpret and compare
the data.

Examples
1. The bar graph shows the number of athletes who represented five
African countries in an international championship.

102 Mathematics | Student Book | Senior Five | Experimental Version

 a) What was the total number of athletes representing the five countries?
b) What was the smallest number of athletes representing one country?
c) What was the most number of athletes representing a country?
d) Represent the information on the graph on a frequency table.

Solution:
We read the data on the graph:
a) Total number of athletes are: 18 + 10 + 22 + 6 + 16 = 72 athletes
b) 6 athletes
c) 22 athletes
d) Representation of the given information on the graph on a frequency
table.

Country Number of athletes

Rwanda 18
Nigeria 10
Tanzania 22
Egypt 6
South Africa 16
Total 72

Mathematics | Student Book | Senior Five | Experimental Version 103

 2. Use a scale vertical scale 2cm: 10 students and Horizontal scale 2cm: 10
represented on histogram below to answers the questions that follows

a) estimate the mode

b) Calculate the range

Solution:
a) To estimate the mode graphically, we identify the bar that represents
the highest frequency. The mass with the highest frequency is 60 kg.
It represents the mode.
b) The highest mass = 67 kg and the lowest mass = 57 kg
Then, The range=highest mass-lowest mass= 67 kg − 57 kg =
10kg

104 Mathematics | Student Book | Senior Five | Experimental Version

 Application activity 4.2.5
The line graph below shows bags of cement produced by CIMERWA
industry cement factory in a minute.

a) Find how many bags of cement will be produced in: 8 minutes, 3

minutes12 seconds, 5 minutes and 7 minutes.
b) Calculate how long it will take to produce: 78 bags of cement.
c) Draw a frequency table to show the number of bags produced
and the time taken.

Mathematics | Student Book | Senior Five | Experimental Version 105

4.3 Numerical descriptive measures
4.3.1 Describing data using mean, median, and mode

Learning Activity 4.3.1

Consider a portfolio that has achieved the following returns: Q1=+10%,
Q2=-3%, Q3=+8%, Q4=+12%, Q5=-7%, Q6=+12% and Q7=+3% over seven
quarters.
a) What is the average return on investment?
b) Which return of the portfolio is in the middle?
c) Which return of the portfolio that has been achieved frequently?

CONTENT SUMMARY
A measure of central tendency is very important tool that refer to the centre
of a histogram or a frequency distribution curve. There are three measures of
central tendency:
• Mean
• Median
• Mode
Difference between the mean, median and mode
• Mean is the average of a data set.
• The median is the middle value in a set of ranked observations. It is also
defined as the middle value in a list of values arranged in either ascending
or descending orders.
• Mode is the most frequently occurring value in a set of values.

How to find mean

The most used measure of central tendency is called mean (or the average).
Here the main of interest is to learn how to calculate the mean when the data
set is raw data.
The following steps are used to calculate the mean:
Step 1: Add the numbers
Step 2: Count how many numbers there are in the data set
Step 3: Find the mean by dividing the sum of the data values by the number of
data values.

106 Mathematics | Student Book | Senior Five | Experimental Version

Mathematically, mean is calculated as follows:

∑x i

x= i =1 , where n is the number of observations in the dataset,

n
xi are observations.

1
Or x =
n
∑ xfi
Here, the mean can also be calculated by multiplying each distinct value by its
frequency and then dividing the sum by the total number of data values.

How to find median

• Rank the given data sets (in increasing or decreasing order)
• Find the middle term for the ranked data set that obtained in step 1.
• The value of this term represents the median.
In general form, calculating the median depends on the number of
observations (even or odd) in the data set, therefore applying the above steps
requires a general formula.

Consider the ranked data x1 , x2 , x3 ,..., xn the formula for calculating the median
for the two cases (even and odd) is given by:

If n is odd, Median = x n +1  , or
 
 2 

th
 n +1 
median is given by   number which is located on this position
 2 
x n  + x n 
   +1
2 2 
If n is even, Median = , or
2

1  n   n  
th th

Median is given by   +  + 1  , then the median is a half of the sum of

2  2   2  
number located on those two positions.

Mathematics | Student Book | Senior Five | Experimental Version 107

Example 1
To understand the three statistical concepts, consider the following example:
A Supermarket recently launched a new mint chocolate chip ice cream flavour.
They want to compare customer traffic numbers to their store in the past seven
days since the launch to understand whether their new offering intrigued
customers. Here is the customer data from last week: Monday =92 customers,
Tuesday =92 customers, Wednesday =121 customers, Thursday =120
customers, Friday = 132 customers, Saturday = 118 customers, and Sunday
=128 customers. To make sense of this data, we can calculate the average:
• Find the sum by adding the customer data together, 92 + 92 + 118 + 120 +
121 + 128 + 132 = 803
• Number of days is equal to 7.

803
• Mean or average is = 114.714 . This means that mean average of
7
customers in the past week is 115 customers.
The mode is 92 customers because on Monday and Tuesday, 92 customers
were received. To find the median, we need to arrange data as follows: 92, 92,
118, 120, 121, 128, 132. Then, the middle value is 120. Therefore, the median
is 120.

By using the formula, n is equal to 7 which is odd. The median = x n +1  is

x 7 +1  = x4 
 2 


 
 2  .
The value at the fourth position in the ranked data above is 120. Hence,
median is 120.

Example 2
Calculate the mean of the pocket money of some 5 students who get
2500 FRW , 4000 FRW , 5500 FRW , 7500 FRW and 3000 FRW .

Sum all the pocket money of five students

= ( 2500 + 4000 + 5500 + 7500 + 3000 ) FRW= 22500 FRW .

=
Divide the sum by the number of students 22500 / 5
= 4500FRW .

The mean of the pocket money of 5 students is 4500FRW .

108 Mathematics | Student Book | Senior Five | Experimental Version

 Application activity 4.3.1
Find the average and median monthly salary (FRW) of all six secretaries
each month earn (in thousands) 104, 340, 140, 185, 270, and 258 each,
respectively.

4.3.2 Summarizing data using variance, standard deviation,

and coefficient of variation

Learning Activity 4.3.2

You and your friends have just measured the heights of your dogs (in
millimeters):

The heights (at the shoulders) are 600mm, 470mm, 170mm, 430mm, and
300mm.
a) Work out the mean height of your dogs.
b) For each height subtract the mean height and square the difference
obtained (the squared difference).
c) Work out the average of those squared differences. What do you
notice about the average?

CONTENT SUMMARY
The following are measures of variation:
• Variance
• Standard deviation
• Range
• Mean deviation.

Mathematics | Student Book | Senior Five | Experimental Version 109

Variance
Variance measures how far a s t of numbers is spread out. A variance of zero
e identical. Variance is always non-negative: a
indicates that all the values are
small variance indicates that the data points tend to be very close to the mean
and hence to each other, while a high variance indicates that the data points are
very spread out around the mean and from each other.
• For the population, the variance is denoted and defined by:

Σ(x − µ)
2

σ =
2

N , where x is individual value, µ is the population mean, and

N is the population size.
• For the sample, the variance is denoted and defined by:
2
 −

Σ x − x
S2 =   −
n − 1 , where x is individual value, x is the sample mean, and n is

the sample size.

How to find variance

To calculate the variance follow these steps:
• Work out the mean (the simple average of the numbers)
• Then for each number: subtract the Mean and square the result (the squared
difference).
• Then work out the average of those squared differences.

Example1:
The heights (in meters) of six children are 1.42, 1.35, 1.37, 1.50, 1.38 and 1.30.
Calculate the mean height and the variance of the heights.
Solution:

1
Mean= (1.42 + 1.35 + 1.37 + 1.50 + 1.38 + 1.30=) 1.39 m
6
1
Variance=
6
(1.422 + 1.352 + 1.37 2 + 1.502 + 1.382 + 1.302 ) − (1.39 )= 0.00386 m
2

110 Mathematics | Student Book | Senior Five | Experimental Version

Example2:
The number of customers served lunch in a restaurant over a period of 60 days
is as follows:

Number of
customers
served 20 − 29 30 − 39 40 − 49 50 − 59 60 − 69 70 − 79
lunch
Number
of days in
6 12 16 14 8 4
the 60-day
period

Find the mean and variance of the number of customers served lunch using this
grouped data.
Solution
To find the mean from grouped data, first we determine the mid-interval values
for all intervals;

Intervals Mid-interval values F r e q u e n c y fi xi fi xi2

(xi) (fi)
20-29 24.5 6 147 3601.5
30-39 34.5 12 414 14283.0
40-49 44.5 16 712 31684.0
50-59 54.5 14 763 41583.5
60-69 64.5 8 516 33282.0
70-79 74.5 4 298 22201.0

∑ = 60 ∑ = 2850 ∑ = 146635
2850
is x
The mean= = 47.5
60
146635
− ( 47.5 ) = 187.67 .
2
Variance =
60

Mathematics | Student Book | Senior Five | Experimental Version 111

 Standard deviation
A most used measure of variation is called standard deviation denoted by ( σ
for the population and S for the sample). The numerical value of this measure
helps us how the values of the dataset corresponding to such measure are
relatively closely around the mean.
Lower value of the standard deviation for a data set, means that the values
are spread over a relatively smaller range around the mean. Larger value of
the standard deviation for a data set means that the values are spread over a
relatively smaller range around the mean.

How to find standard deviation

Take a square root of the variance. The population standard deviation is defined
as: square root of the average of the squared differences from the population
mean.

Σ(x − µ)
2

σ =
= σ 2
, where x is individual value, µ is the population
N
mean, and N is the population size.
The sample standard deviation is defined as: square root of the average of
the squared differences from the sample mean.
2
 −

Σ x − x −

=S S2
=   , where x is individual value, x is the sample mean,
n −1
and n is the sample size.

Example:
The six runners in a 200 meter race clocked times (in seconds) of 24.2, 23.7,
25.0, 23.7, 24.0, 24.6.
Find the mean and standard deviation of these times.

Solution

24.2 + 23.7 + 25.0 + 23.7 + 24.0 + 24.6

=x = 24.2 seconds
6

112 Mathematics | Student Book | Senior Five | Experimental Version

 ( 24.2 − 24.2 ) + ( 23.7 − 24.2 ) + ( 25.0 − 24.2 ) + ( 23.7 − 24.2 ) + ( 24.0 − 24.2 ) + ( 24.6 − 24.2 )
2 2 2 2 2 2

σ=
6
= 0.473 seconds

Range
The range for a data set is depends on two values (the smallest and the largest
values) among all values in such data set. The range is defined as the difference
between the largest value and the lowest value.

Mean deviation
Another measure of variation is called mean deviation; it is the mean of the
distances between each value and the mean.

Coefficient of variation
A coefficient of variation (CV) is one of well-known measures that used to
compare the variability of two different data sets that have different units of
measurement.
Moreover, one disadvantage of the standard deviation that its being a measure
of absolute variability and not of relative variability.
The coefficient of variation, denoted by (CV), expresses standard deviation as a
percentage of the mean and is computed as follows:

σ
For population data CV= ×100%
µ

S
For sample data CV= −
×100%
x
Example:
Two plants C and D of a factory show the following results about the number of
workers and the wages paid to them.

C D
No. of workers 5000 6000
Average monthly wages $2500 $2500
Standard deviation 9 10

Mathematics | Student Book | Senior Five | Experimental Version 113

Using coefficient of variation, find in which plant, C or D there is greater
variability in individual wages.
In which plant would you prefer to invest in?

Solution
To find which plant has greater variability, we need to find the coefficient of variation.
The plant that has a higher coefficient of variation will have greater variability.
Coefficient of variation for plant C:
Using coefficient of variation formula,
σ
C.V . = ×100%, x ≠ 0
x

9
C.V . = ×100% = 0.36%
2500

Now, Coefficient of variation for plant D:

10
C.V . = ×100% = 0.4%
2500

Plant C has CV = 0.36 and plant D has CV = 0.4

Hence plant D has greater variability in individual wages.
I would prefer to invest in plant C as it has lower coefficient (of variation)
because it provides the most optimal risk-to-reward ratio with low volatility
but high returns.

Application activity 4.3.2

A music school has budgeted to purchase three musical instruments. They
plan to purchase a piano costing $3,000, a guitar costing $550, and a drum
set costing $600. The mean cost for a piano is $4,000 with a standard
deviation of $2,500. The mean cost for a guitar is $500 with a standard
deviation of $200. The mean cost for drums is $700 with a standard
deviation of $100. Which cost is the lowest, when compared to other
instruments of the same type? Which cost is the highest when compared
to other instruments of the same type. Justify your answer.

114 Mathematics | Student Book | Senior Five | Experimental Version

 4.3.3 Determining the position of data value using quartiles

Learning Activity 4.3.3

Consider the prices of 11 items arranged in order of rank in the table below.

Rank
1 2 3 4 5 6 7 8 9 10 11
order

Prices
4500 4600 4800 5200 5400 5600 6300 6400 7700 7700 7800
(FRW)

i) Identify the median price.

ii) How many items were bought at a price below the median price?
iii) How many items were bought at a price above the median price?
iv) Find the middle price of the lower half of the set of prices.
v) Find the middle price of the upper half of the set of prices.
vi)Together with the median price, what do the middle prices in (iv) and
(v) do to the given data? Discuss.

CONTENT SUMMARY
Any data set can be divided into four equal parts by using a summary measure
called quartiles.
There are three quartiles that used to divide the data set which is denoted by Qi
for i = 1, 2,3 . The following definition is illustrated the meaning of the quartiles.
Quartiles are three summary measures that divide a ranked data set into four
equal parts. The following are three quartiles:
• First quartile ( Q1 ) is the middle term among the observations that are less
than the median.
• Second quartile ( Q2 ) is the sa e as the median.
m
• Third quartile ( Q3 ) is the value of the middle term among the observations
that are greater than the median.

Mathematics | Student Book | Senior Five | Experimental Version 115

How to find quartiles
th
1 
 ( n + 1) 
In general, the lower quartile, Q1 takes the  4  position from the lower

th
3 
end on the rank order. The upper quartile, Q3 takes the  ( n + 1)  position on
4 
th th
1  3 
the rank order. For large population, it is enough to use  (n)  and  ( n ) 
positions for the lower and upper quartiles respectively. 4  4 

Example
For the given data set 61, 24, 39, 51, 37, 59, 45. Find the values of the three
quartiles. First, we rank the given data in increasing order. Then we calculate
the three quartiles as follows 24 37 39 45 51 59 61.
• Median value is 45.
• Values less than the median are 24 37 and 39.
• Values greater than the median are 51 59 and 61.
Therefore, the values of the three quartiles are Q1 = 37 , Q2 = 45 Q3 = 59 .

Application activity 4.3.3

The heights in cm of 13 boys are: 163, 162, 170, 161, 165, 163, 162, 163,
164, 160, 158, 153, 165. Determine the three quartiles.

4.5 Measure of symmetry

4.5.1 Skewness

Learning Activity 4.3.4

Consider the two data sets that were recorded for the temperature:
Dataset1: 78, 78, 79,77,76,72,74,75,74,75,76,77, 76; Dataset2: 66, 65, 58,
59, 61, 59, 61, 58, 60, 64, 59, 64, 60, 59, 58, 59, 61, 58, 60, 61, 58, 60, 63, 58,
60, 63, 58, 60, 63, 59.
a) Represent the two datasets using bar graphs.
b) Find the mean, median, and mode of each dataset.
c) Compare the mean, median, and mode of each dataset. Do all
measures of a central tendency (mean, median, and mode) lie in
the middle?

116 Mathematics | Student Book | Senior Five | Experimental Version

CONTENT SUMMARY
Sometimes data are distributed equally on the right and left of the mean value.
Such data are said to be normally distributed or bell curved. They are also called
symmetric. This means that the right and the left of the distribution are perfect
mirror images of one another.
Not all data is symmetrically distributed. Sets of data that are not symmetric are
said to be asymmetric. The measure of how data are asymmetric or symmetric
can be is called skewness. The mean, median and mode are all measures of the
center of a set of data. The skewness of the data can be determined by how
these quantities are related to one another. There are three types of skewness:
• Right skewness
• Zero skewness
• Left skewness.

Figure 1: Three types of skewness (Positive skew, Zero skew or symmetrical distribution, and
Negative skew)

Data skewed to the right (Positive skewness).

Data that are skewed to the right have a long tail that extends to the right. An
alternate way of talking about a data set skewed to the right is to say that it is
positively skewed. Generally, most of the time for data skewed to the right, the
mean will be greater than the median and both are greater than the mode.
In summary, for a data set skewed to the right: Mode > Median >Mode.

Data skewed to the left (Negative skewness).

Data that are skewed to the left have a long tail that extends to the left. An
alternate way of talking about a data set skewed to the left is to say that it is
negatively skewed. Generally, most of the time for data skewed to the left, the
mean will be less than the median and both less than the mode.

Mathematics | Student Book | Senior Five | Experimental Version 117

In summary, for a data set skewed to the left: Mode > Median >Mode.

Zero skewness
The symmetrical data has zero skewness as all measures of a central tendency
lies in the middle.   
Mode Median
In summary, for a data set skewed to the left:= = Mean .

Measure of skewness
It’s one thing to look at two sets of data and determine that one is symmetric
while the other is asymmetric. It’s another to look at two sets of asymmetric
data and say that one is more skewed than the other. It can be very subjective
to determine which is more skewed by simply looking at the graph of the
distribution. This is why there are ways to numerically calculate the measure
of skewness.
One measure of skewness, called Pearson’s first coefficient of skewness, is
to subtract the mean from the mode, and then divide this difference by the
standard deviation of the data.

Mean − Mode
Skewness =
σ
The reason for dividing the difference is so that we have a dimensionless
quantity. This explains why data skewed to the right has positive skewness.
If the data set is skewed to the right, the mean is greater than the mode, and
so subtracting the mode from the mean gives a positive number. A similar
argument explains why data skewed to the left has negative skewness.
Pearson’s second coefficient of skewness is also used to measure the asymmetry
of a data set. For this quantity, we subtract the mode from the median, multiply
this number by three and then divide by the standard deviation.

3 ( Median − Mode )
Skewness =
σ
Examples in real life
Incomes are skewed to the right because even just a few individuals who
earn millions of dollars can greatly affect the mean, and there are no negative
incomes.
Data involving the lifetime of a product, such as a brand of light bulb, are skewed
to the right. Here the smallest that a lifetime can be is zero, and long lasting
light bulbs will impart a positive skewness to the data.

118 Mathematics | Student Book | Senior Five | Experimental Version

 Application activity 4.3.3
Discuss the skewness of the data represented in the histograms below.

4.5.2 Chebyshev’s theorem and Empirical rule

Learning Activity 4.5.2

The prices of ten items in the supermarket are as follows:

Items 1 2 3 4 5 6 7 8 9 10

Price (in thousands (FRW)) 75 74 75 72 73 72 73 74 72 74

a) Find the mean price and the standard deviation.

b) How many items with prices falling within one standard deviation
from the mean?
c) How many items with prices falling within two standard deviations
from the mean?
d) How many items with prices falling within three standard
deviations from the mean?

Mathematics | Student Book | Senior Five | Experimental Version 119

CONTENT SUMMARY
There are two ways in which we can use the sta dard deviation to make a
n
statement regarding the proportion o measurements that fall within various
f
intervals of values centered at the mean value. The information depends on the
shape of histogram.
• If the histogram is bell shaped (symmetric data), the Empirical Rule is used.
• Other
wise (If data are asymmetric), Chebyshev’s theorem is used.

The Empirical Rule

The Empirical Rule makes more precise statements, but it can be applied
only to symmetric data (normally distributed data). For such a sample of
measurements, the Empirical Rule states that:

• Approximately 68% of the observations fall in the interval

[µ − σ , µ + σ ] ,
• Approximately 95% of the observations fall in the interval
[ µ − 2σ , µ + 2σ ]
• Approximately 99.7% of the observation fall in the interval
[ µ − 3σ , µ + 3σ ]

Chebyshev’s theorem
Chebyshev’s theorem which applies to any set of measurements (all shapes of
histograms). It states that the proportion of observations
1 that lie within k
1 − 2 , where k > 1 .
standard deviations of the mean is at least k
When k=2, the Chebyshev’s theorem states that at least three-quarters (75%)
of all observations lie within two standard deviations of the mean.
With k=3, Chebyshev’s theorem states that at least eight-ninths (88.9%) of all
observations lie within three standard deviations of the mean.

120 Mathematics | Student Book | Senior Five | Experimental Version

 Application activity 4.5.2
Sweets are packed into bags with a normal mass of 75g. Ten bags are
picked at random from the production line and weighed.
Their masses in grams are 76, 74.2, 75.1, 73.7, 72, 74.3, 75.4, 74, 73.1, and
72.8.
a) Find the mean mass and the standard deviation. It was later
discovered that the scale was reading 3.2g below the correct
weight.
b) What was the correct mean mass of the ten bags and the correct
standard deviation?
c) Compare your answer in a) and b) and comment.

4.6 Examples of applications of univariate statistics

in mathematical problems that involve finance,
accounting, and economics.

Learning Activity 4.6

Referring to the concepts you have learnt in this unit, list down concepts
and give examples of how those concepts are applied in solving problems
related to finance, accounting, and economics.

CONTENT SUMMARY
With the use of descriptive statistics, we can summarize data related to revenue,
expenses, and profit for companies. For example, a financial analyst who works
for a retail company may calculate the following descriptive statistics during
one business quarter:
• Mean and median number of daily sales
• Standard deviation of daily sales
• Total revenue and total expenses
• Percentage change in new customers
• Percentage of products returned by customers.
• The mean household income.
• The standard deviation of household incomes.
• The sum of gross domestic product.

Mathematics | Student Book | Senior Five | Experimental Version 121

 • The percentage change in total new jobs.
Using these metrics, the analyst can gain a strong understanding of the current
financial state of the company and also compare these metrics to previous
quarters to understand how the metrics are trending over time. Analyst can
then use these metrics to inform the organization on areas that could use
improvement to help the company increases revenue or reduce expenses.

Application activity 4.6

1. As the controller of the XXX Corporation, you are directed by the
board’s chairman to investigate the problem of overspending
by employees with expense accounts. You ask the accounting
department to provide records of the number of FRW spent by each
of 25 top employees during the past month. The following record
is provided: 292000, 494000, 600000, 807000, 535000, 435000,
870000, 725000, 299000, 602000, 322000, 397000, 390000,
420000, 469000, 712000, 520000, 575000, 670000, 723000,
560000, 298000, 472000, 905000, 305000. The questions the
board of directors wanted to be answered are:
a) How many of our 25 top executives spent more than 600000FRW
last month?
b) On average, how much do employees spend?
2. Consider the data on household size, annual income (in thousands
(FRW)), and the number of cows for each household.

Annual
370 490 580 680 610 640 790 890 980 950
income
Household
2 4 4 1 3 5 6 4 7 2
size
Number of
0 0 1 3 2 2 1 1 1 0
cows

Estimate the average annual income.

122 Mathematics | Student Book | Senior Five | Experimental Version

 End of unit assessment 4

1. You are assigned by your general manager to examine each of last

month’s sales transactions. Find their average, find the difference
between the highest and lowest sates figures, and construct a
chart showing the differences between charge account and cash
customers. Is this a problem in descriptive or inferential statistics?
2. When a cosmetic manufacturer tests the market to determine
how many women will buy eyeliner that has been tested for safety
without subjecting animals to injury, is it involved in a descriptive
statistics problem or an inferential statistics problem? Explain your
answer.
3. Suppose a real estate broker, is interested in the average price of a
home in a development comprising 100 homes.
a) If she uses 12 homes to predict the average price of all 100 homes,
is she using inferential or descriptive statistics?
b) If she uses all 100 homes, is she using inferential or descriptive
statistics?
4. The number of sales a salesman had in the previous 7 days are: 5, 1,
2, 1, 6, 5, and 1. Calculate the variance and standard deviation.
5. Five people waiting in line at a bank were randomly chosen and
asked how much cash they had in their pocket. The amounts in
dollars are: 16, 17, 18, 19, and 15. Find the variance and standard
deviation.
6. Refer to the table below, in which annual macroeconomic data
including GDP, CPI, prime rate, private consumption, private
investment, net exports, and government expenditures from 1995
to 2000 are given. Answer the following questions.
a) How many observations are in the data set?
b) How many variables are in the data set?
c) Which of the variables are qualitative and which are quantitative
variables?

Mathematics | Student Book | Senior Five | Experimental Version 123

 Govern-
Private Net
Private ment
GPD consump- exports
Prime investment expendi-
Year (millions CPI tion
rate (Mil- tures
of FRW) (Millions of
(Millions lions of
FRW) (Millions
of FRW) FRW)
of FRW)
1995 230.9 29.585 4.83 178.4 296.5 111.5 111.5
1996 296.9 29.902 4.50 182.2 294.6 119.5 119.5
1997 302.4 30.253 4.50 191.2 332.0 130.1 130.1
1998 326.7 30.633 4.50 198.9 354.3 136.4 136.4
1999 332.3 31.038 4.50 210.4 383.5 143.2 143.2
2000 3610.1 31.528 4.54 2241.8 437.3 151.4 151.4

124 Mathematics | Student Book | Senior Five | Experimental Version

 Unit 5 Bivariate statistics and
Applications

Key unit competence: Apply bivariate statistical concepts to collect, organise,

analyse, present, and interpret data to draw appropriate
decisions.

Introductory activity

In Kabeza village, after her 9 observations about farming, UMULISA saw

that in every house observed, where there is a cow (X) if there is also
domestic duck (Y), then she got the following results:

(1, 4 ) , ( 2,8) , ( 3, 4 ) , ( 4,12 ) , ( 5,10 )

( 6,14 ) , ( 7,16 ) , (8, 6 ) , ( 9,18)

a) Represent this information graphically in ( x, y ) − coordinates .

b) Find the equation of line joining any two point of the graph and
guess the name of this line.
c) According to your observation from (a), explain in your own
words if there is any relationship between Cows (X) and
domestic duck (Y).

Mathematics | Student Book | Senior Five | Experimental Version 125

5.1 Introduction to bivariate statistics.
5.1.1. Key concepts of bivariate statistics

Learning Activity 5.1.1

Here is a set of data relating the temperature on days in July and the number
of ice creams sold in a corner shop.

Temperature (° C) 14 16 15 16 23 12 21 22
Ice cream sales 16 18 14 19 43 12 24 26
If a businessman wants to make future investments based on how ice cream
sales relate to the temperature,
i) Which statistical measure and variables will he use?
ii) Which variable depends on the other?
iii) In statistics, how do we call a variable which depends on the other?

iv) Plot the corresponding points ( x, y ) on a Cartesian plane and describe

the resulting graph. Can you draw a line roughly representing the
points in your graph?
v) Can you obtain a rule relating x and y ? Explain your answer.

CONTENT SUMMARY
Bivariate data is data that has been collected in two variables, and each data
point in one variable has a corresponding data point in the other value. Bivariate
dataisobservationontwovariables,whilstunivariatedataisanobservationon
only one variable. We normally collect bivariate data to try and investigate the
relationshipbetweenthetwovariablesandthenusethisrelationshiptoinform
future decisions.
Bivariate statistics deals with the collection, organization, analysis,
interpretation, and drawing of conclusions from bivariate data. Data sets that
contain two variables, such as wage and gender, and consumer price index and
inflationratedataaresaidtobebivariate.Inthecaseofbivariateormultivariate
data sets we are often interested in whether elements that have high values of
one of the variables also have high values of other variables.
In bivariate statistics, we have independent variable and dependent variable.
Dependent variable refers to variable that depends on the other variable (s).
Independent variable refers to variable that affects the other variable (s).

126 Mathematics | Student Book | Senior Five | Experimental Version

Scatter diagram is the graph that represents the bivariate data in x and y
cartesian plane. The points do not lie on a line or a curve, hence the name. A
scatter graph of bivariate data is a two-dimensional graph with one variable on
one axis, and the other variable on the other axis. We then plot the corresponding
points on the graph. We can then draw a regression line (also known as a line of
best fit), and look at the correlation of the data (which direction the data goes,
and how close to the line of best fit the data points are).For example, the scatter
diagram showing the relationship between secondary school graduate rate and
the % of residents who live below the poverty line.

Figure 5.1: Scatter diagram representing the relationship between secondary school graduate
rate and the poverty.

Examples of bivariate statistics

• Collecting the monthly savings and number of family members’ data
of every family that constitutes your population if you are interested in
finding the relationship between savings and number of family members.
In this case, you will take a small sample of families from across the country
to represent the larger population of Rwanda. You will use this sample to
collect data on family monthly savings and number of family members.
• We can collect data of outside temperature versus ice cream sales, these
would both be examples of bivariate data. If there is a relationship showing
an increase of outside temperature increased ice cream sales, then shops
could use this information to buy more ice cream for hotter spells during
the summer.

Mathematics | Student Book | Senior Five | Experimental Version 127

 Application activity 5.1.1
1. Using an example, differentiate univariate statistics from bivariate
statistics.
2. Using an example, differentiate dependent variable from
independent variable.

5.2 Measures of linear relationship between two variables:

covariance, Correlation, regression line and analysis,
and spearman’s coefficient of correlation.
5.2.1 Covariance and correlation.

Learning Activity 5.2.1

As students of economics, you might be interested in whether people with
more years of schooling earn higher incomes. Suppose you obtain the data
from one district for the population of all that district households. The data
contain two variables, household income (measured in FRW) and a number
of years of education of the head of each household.
i) Which statistical measure will you use to know whether people with
more years of schooling earn higher incomes.
ii) If you want to know how household income and year of schooling
covariate, which statistical measure will you consider?

CONTENT SUMMARY
Covariance is a statistical measure that describes the relationship between a
pair of random variables where change in one variable causes change in another
variable. It takes any value between -infinity to + infinity, where the negative
value represents the negative relationship whereas a positive value represents
the positive relationship. It is used for the linear relationship between variables.
It gives the direction of relationship between variables.

How to calculate the covariance

From learning activity 5.2.1, let xi be the value of annual household income for
household i and yi be the number of years of schooling of the head of the i th
household. Now consider a random sample of n households which yields the
paired observations ( xi , yi ) for i = 1, 2,3,..., n.

128 Mathematics | Student Book | Senior Five | Experimental Version

The covariance of annual household income and schooling years from the
learning activity 5.2.1 is given by

1 n  −
 −
 1 n − −
cov(x, y)= ∑ i  i 

n i =1 
x − x   y − y  or cov(x,
= y) ∑ i i y.
n i =1
x y − x
The covariance of variables x and y is a measure of how these two variables
change together. If the greater values of one variable mainly correspond with the
greater values of the other variable, and the same holds for the smaller values,
i.e., the variables tend to show similar behavior, the covariance is positive. In
the opposite case, when the greater values of one variable mainly correspond
to the smaller values of the other, i.e., the variables tend to show opposite
behavior, the covariance is negative. If covariance is zero, the variables are said
to be uncorrelated, meaning that there is no linear relationship between them.

Correlation
The Pearson’s coefficient of correlation (or product moment coefficient of
correlation or simply coefficient of correlation), denoted by r , is a measure
of the strength of linear relationship between two variables. The coefficient

cov ( x, y )
of correlation between two variables x and y is given by r = , where
σ xσ y
cov ( x, y ) is covariance of x and y , σ x is the standard deviation for x , σ y is
the standard deviation for y . Correlation describes how the two variables are
related. The correlation coefficient ranges from -1 to +1.
There are three types of correlation:
• Negative correlation
• Zero correlation
• Positive correlation
If the linear coefficient of correlation takes values closer to -1, the correlation is
strong and negative, and will become stronger the closer r approaches -1.
If the linear coefficient of correlation takes values close to 1, the correlation is
strong and positive, and will become stronger the closer r approaches 1. If the
linear coefficient of correlation takes values close to 0, the correlation is weak.
If r = 1 or r=-1, there is perfect correlation and the line on the scatter plot is
increasing or decreasing respectively. If r = 0, there is no linear correlation.

Mathematics | Student Book | Senior Five | Experimental Version 129

 Application activity 5.1.1
The production manager had ten newly recruited workers under him. For
one week, he kept a record of the number of times that each employee
needed help with a task and make a scatter diagram for the data. What
type of correlation is there?
Employee A B C D E F G H I J
Length of 24 11 47 58 3 70 76 44 33 87
employment (weeks)
Request for help 14 20 10 13 25 16 6 15 19 6

5.2.2 Regression line and analysis

Learning Activity 5.2.2

In a village, Emmanuella visited nine families and their farming
activities. For each visited family there were x number of cows,
and y goats. Emmanuella recorded her observations as follows:
(1, 4 ) , ( 2,8) , ( 3, 4 ) , ( 4,12 ) , ( 5,10 ) , ( 6,14 ) , ( 7,16 ) , (8, 6 ) , ( 9,18) .
a) Represent Emmanuella’s recorded observations graphically in
cartesian plane.
b) Connect any two points on the graph drawn above to form a
straight line and find equation of that line. How are the positions
of the non-connected points vis-à-vis that line?
c) According to your observation from a., is there any relationship
between the variation of the number of cows and the number of
goats? Explain.

130 Mathematics | Student Book | Senior Five | Experimental Version

CONTENT SUMMARY
Bivariate statistics can help in prediction of a value for one variable if we know
the value of the other. We use the regression line to predict a value of y for any
given value of x and vice versa. The “best” line wou d ma e the best predictions:
l from
the observed y-values should stray as little as possible k the line. This straight
line is the regression line from which we can adjust its algebraic expressions
and it is written as =y ax + b .

Deriving regression line equation

The regression line y on x has the form = y ax + b . We need the distance from
this line to each point of the given data to be small, so that the sum of the square
of such distances be very small.
n n

∑  yi − ( axi + b )
2
∑ ( y − ax − b )
2
D= D= i i
That is i =1 or i =1 (1) is minimum.
1. Differentiate relation (1) with respect to b . In this case, x, y and a will
be considered as constants.
2. Equate relation obtained in 1 to zero, divide each side by n and give the
value of b .
• Take the value of b obtained in 2 and put it in relation obtained in 1.
Differentiate the obtained relation with respect to Variance for variable x

2
1 n 
∑  i 
−
is σ x2
= x − x
n i =1 
2
1 n  −

y is σ y2
• Variance for variable = ∑  yi − y 
n i =1  
1 n  −
 −

• Covariance of these two variables is cov ( x, y )= ∑  xi − x   yi − y  , give
n i =1   
the simplified expression equal to a.
3. a , equate it to zero and divide both sides by n to find the value of a .
4. Using the relations in 2:

5. Put the value of b obtained in 2 and the value of a obtained in 4 in relation

y ax + b and give the expression of regression line y on x . Hence, the
=

cov ( x, y )  − cov ( x, y ) − 
regression line y on x is written
= as y x +  y− x .
σ 2
x  σ 2
x 

Mathematics | Student Book | Senior Five | Experimental Version 131

 cov ( x, y ) 
− −

We may write it as Ly=
/x ≡ y − y  x − x .
σx 2
 
Note that the regression line x on y is = x cy + d given by

− cov ( x, y )  −

x−x
=  y − y .
σ y2  

− cov ( x, y )  −

This line is written as Lx=
/y ≡ x − x  y − y .
σ y2  
Example
Find the regression line of y on x for the following data and estimate the value
of y for x = 4 , x = 7 , x = 16 and the value of x for y = 7 , y = 9 , y = 16 .
x 3 5 6 8 9 11
y 2 3 4 6 5 8

Answer:
x y − − 2 2
x − x y − y  x − x      
− − − −

i y
 i − y   xi − x   yi − y 
      
3 2 -4 -2.6 16 6.76 10.4
5 3 -2 -1.6 4 2.56 3.2
6 4 -1 -0.6 1 0.36 0.6
8 6 1 1.4 1 1.96 1.4
9 5 2 0.4 4 0.16 0.8
11 8 4 3.4 16 11.56 13.6
6 6 6 2 2
 
∑ xi = 42
− 6

∑ yi = 28 ∑ ∑  y − y 
− 6

∑  x − x   y − y  =

− −
42
 xi − x  = 23.36
= i i 30
i =1   i
i =1 i =1 i =1 i =1

− 42 − 28
x
= = 7 ,=
y = 4.7 ,
6 6

1 6  −
 −
 30
cov ( x, y ) = ∑  xi − x   yi − y  = , = 5
6 i =1    6
2
1 6  −
 42
σ x2 = ∑ x
 i
6 i =1 
− x  = =7 ,
 6
132 Mathematics | Student Book | Senior Five | Experimental Version
 2
1 6  23.36
σ = ∑  yi − y  =
−
2
y = 3.89.
6 i =1   6

5
The regression line of y on x is Ly / x ≡ y − 4.7= ( x − 7) .
7

Application activity 5.1.1

Consider the following data:

x 60 61 62 63 65

y 3.1 3.6 3.8 4 4.1

Find the regression line of y on x and deduce the approximated value of y

when x=64.

5.2.3 Spearman’s coefficient of correlation

Learning Activity 5.2.2

The death rate data from 1995 to 2000 for developed and underdeveloped
countries are displayed in the table below.
The death rate for developed 2 8 4 6 5 3
countries (y)
The death for underdeveloped 3 11 6 8 9 5
countries (x)

i) Write the death rate for developed countries and the death rate for
underdeveloped countries in ascending order.
ii) Rank the death rate for developed countries and the death rate for
underdeveloped countries such that the lowest death rate is ranked 1
and the highest death rate is ranked 6.

Mathematics | Student Book | Senior Five | Experimental Version 133

 iii) Complete the table below using the information above.

Year x y R a n k R a n k Rank (x)- Rank (y)=d d2

(x) (y)
1995
1996
1997
1998
1999
2000
n=6 n

∑d
i =1
i
2
=?

n
6∑ di2
iv) Using the information completed in the table above, find 1 − i =1

n ( n − 1)
2

CONTENT SUMMARY
A Spearman coefficient of rank correlation or Spearman’s rho is a measure
of statistical dependence between two variables. It assesses how well the
relationship between two variables can be described using a monotonic
n
function.6Thed Spearman’s coefficient of rank correlation is denoted and defined
by
∑ i
2

ρ = 1 − i =12
n ( n − 1)
, where, d refers to the difference of ranks between paired
items in two series and n is the number of observations.
It is much easier to calculate the Spearman’s coefficient of rank correlation than
to calculate the Pearson’s coefficient of correlation as there is far less working
involved. However, in general, the Pearson’s coefficient of correlation is a more
accurate measure of correlation.

134 Mathematics | Student Book | Senior Five | Experimental Version

 Application activity 5.2.3
Calculate Spearman’s coefficient of rank correlation for the series.

Income (X in 12 8 16 12 7 10 12 16 12 9
thousands
(FRW))
Expenditure (Y 6 5 7 7 4 6 8 13 10 10
in thousands
(FRW))

5.2.4 Application of bivariate statistics

Learning Activity 5.2.4

Referring to what you have learnt in this unit, discuss how bivariate
statistics is used in our daily life.

CONTENT SUMMARY
Bivariate statistics can help in prediction of the value for one variable if we know
the value of the other. Bivariate data occur all the time in real-world situations
and we typically use the following methods to analyze the bivariate data:
• Scatter plots
• Correlation Coefficients
• Simple Linear Regression
The following examples show different scenarios where bivariate data appears
in real life.
Businesses often collect bivariate data about total money spent on advertising
and total revenue. For example, a business may collect the following data for 12
consecutive sales quarters:

Mathematics | Student Book | Senior Five | Experimental Version 135

This is an example of bivariate data because it contains information on exactly
two variables: advertising spend and total revenue. The business may decide to
fit a simple linear regression model to this dataset and find the following fitted
model:
Total Revenue = 14,942.75 + 2.70*(Advertising Spend). This tells the business
that for each additional dollar spent on advertising, total revenue increases by
an average of $2.70.
• Economists often collect bivariate data to understand the relationship
between two socioeconomic variables. For example, an economist may
collect data on the total years of schooling and total annual income among
individuals in a certain city:

136 Mathematics | Student Book | Senior Five | Experimental Version

He may then decide to fit the following simple linear regression model:
Annual Income = -45,353 + 7,120*(Years of Schooling). This tells the economist
that for each additional year of schooling, annual income increases by $7,120
on average.
• Biologists often collect bivariate data to understand how two variables are
related among plants or animals. For example, a biologist may collect data
on total rainfall and total number of plants in different regions:

Mathematics | Student Book | Senior Five | Experimental Version 137

The biologist may then decide to calculate the correlation between the two
variables and find it to be 0.926. This indicates that there is a strong positive
correlation between the two variables. That is, higher rainfall is closely
associated with an increased number of plants in a region.
• Researchers often collect bivariate data to understand what variables
affect the performance of students.
For example, a researcher may collect
data on the number of hours studied per week and the corresponding GPA
for students in a certain class:

She may then create a simple scatter plot to visualize the relationship between
these two variables:

Clearly there is a positive association between the two variables: As the

number of hours studied per week increases, the GPA of the student tends to
increase as well.
138 Mathematics | Student Book | Senior Five | Experimental Version
With the use of bivariate data we can quantify the relationship between
variables related to promotions, advertising, sales, and other variables.
Time series forecasting allows financial analysts to predict future revenue,
expenses, new customers, sales, etc. for a variety of companies.

Application activity 5.2.3

A company is to replace its fleet of cars. Eight possible models are
considered and the transport manager is asked to rank them, from 1 to 8,
in order of preference. A saleswoman is asked to use each type of car for
a week and grade them according to their suitability for the job (A-very
suitable to E-unsuitable). The price is also recorded:
Model S T U V W X Y Z
Transport manager’s 5 1 7 2 8 6 4 3
ranking
Saleswoman’s grade B B+ D- C B+ D C+ A-

Price (£10s) 611 811 591 792 520 573 683 716
a) Calculate the Spearman’s coefficient of rank correlation between:
i) Price and transport manager’s rankings,
ii) Price and saleswoman’s grades.
b) Based on the result of a, state, giving a reason, whether it would
be necessary to use all the three different methods of assessing
the cars.
c) A new employee is asked to collect further data and to do some
calculations. He produces the following results: The coefficient of
correlation between
i) Price and boot capacity is 1.2,
ii) Maximum speed and fuel consumption in miles per gallons is -0.7,
iii) Price and engine capacity is -0.9. For each of his results, say giving
a reason, whether you think it is reasonable.
d) Suggest two sets of circumstances where Spearman’s coefficient
of rank correlation would be preferred to the Pearson’s coefficient
of correlation as a measure of association.

Mathematics | Student Book | Senior Five | Experimental Version 139

 End of unit assessment 5

1. Table below shows the marks awarded to six students in accounting

competition:

Student A B C D E F
Judge1 6.8 7.3 8.1 9.8 7.1 9.2
Judge2 7.8 9.4 7.9 9.6 8.9 6.9

Calculate a coefficient of rank correlation.

2. At the end of a season, a league of eight hockey clubs produced the
following table showing the position of each club in the league and
the average attendance (in hundreds) at home matches.

Club A B C D E F G H
Position 1 2 3 4 5 6 7 8
Average position 27 29 9 16 24 15 12 22

Calculate Spearman’s coefficient of rank correlation between the position

in the league and average attendance. Comment on your results.
3. The following results were obtained from lineups in Accounting
and Finance examinations:

Accounting (x) Finance (y)

Mean 475 39.5
Standard deviation 16.8 10.8
r 0.95

Find both equations of regression lines. Also estimate the value of y for
x=30.

140 Mathematics | Student Book | Senior Five | Experimental Version

 4. The following results were obtained from records of age (x) and
systolic blood pressure (y) of a group of 10 men:

(x) (y)
Mean 53 142
Variance 130 165
n
 −
 −

∑
i =1   
1220
 xi − x   yi − y  =
Find both equations of the regression lines. Also estimate the blood
pressure of a man whose age is 45.

Mathematics | Student Book | Senior Five | Experimental Version 141

REFERENCES
1. Arem, C. (2006). Systems and Matrices. In C. A. DeMeulemeester, Systems
and Matrices (pp. 567-630). Demana: Brooks/Cole Publishing 1993 &
Addison-Wesley1994.
2. Kirch, W. (Ed.). (2008). Level of MeasurementLevel of measurement BT
- Encyclopedia of Public Health (pp. 851–852). Springer Netherlands.
https://doi.org/10.1007/978-1-4020-5614-7_1971
3. Crossman, Ashley. (2020). Understanding Levels and Scales of
Measurement in Sociology. Retrieved from https://www.thoughtco.
com/levels-of-measurement-3026703
4. Markov. ( 2006). Matrix Algebra and Applications. In Matrix Algebra and
Applications (pp. p173-208.).
5. REB, R.E (2020). Mathematics for TTCs Year 2 Social Studies Education.
Student’s book.
6. REB, R.E (2020). Mathematics for TTCs Year 3 Social Studies Education.
Student’s book
7. REB, R.E (2020). Mathematics for TTCs Year 1 Science Mathematics
Education. Students’ book
8. REB, R.E (2020). Mathematics for TTCs Year 1 Social Studies Education.
Student’s book
9. Rossar, M. (1993, 2003). Basic Mathematics For Economists. London and
New York: Taylor & Francis Group.

142 Mathematics | Student Book | Senior Five | Experimental Version